# On the Homogenization of Second Order Level Set PDE in Periodic Media

Peter S. Morfe

###### Abstract.

This paper analyzes two classes of second order level set PDE in periodic
media in the parabolic scaling. First, we study fully nonlinear geometric
operators under general assumptions in dimension $d=2$ and prove that the
associated equations homogenize in this case. Next, a class of quasi-linear
geometric operators is treated in arbitrary dimensions $d\geq 2$. Here we
prove similar results implying homogenization of sub- and supersolutions.
However, by adapting arguments from the study of oscillating boundary value
problems, we show that the homogenized coefficients are generically
discontinuous in all dimensions $d\geq 3$. Finally, we investigate the
connection between the effective mobility obtained in the quasilinear case and
linear response.

##  1\. Introduction

In this paper, we are interested in the behavior, as $\epsilon\to 0^{+}$, of
the solutions of the parabolically scaled, level set PDE

(1) $\left\\{\begin{array}[]{r
l}u^{\epsilon}_{t}-F(Du^{\epsilon},D^{2}u^{\epsilon},\epsilon^{-1}x)=0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty),\\\
u^{\epsilon}=u_{0}&\text{on}\,\,\mathbb{R}^{d}\times\\{0\\}.\end{array}\right.$

Here $F$ is a spatially periodic, geometric operator that has the same
ellipticity properties as the mean curvature operator (see equation (7) for a
precise statement).

When $d=2$ and $F$ is a general geometric operator (see the assumptions in
Section 2.1), we prove that (1) homogenizes to the solution of an effective
equation with nonlinearity $\overline{F}=\overline{F}(p,X)$. In this case, due
to the simpler geometry when $d=2$, $\overline{F}$ is continuous in
$(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}$.

In dimensions $d\geq 3$, we study quasilinear equations and prove that the
effective coefficients describing the large-scale behavior are generically
discontinuous. Specifically, these equations take the form

(2) $\hskip 14.45377pt\left\\{\begin{array}[]{r
l}m(\epsilon^{-1}x,\widehat{Du^{\epsilon}})u^{\epsilon}_{t}-\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du^{\epsilon}})D^{2}u^{\epsilon}\right)=0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty),\\\
u^{\epsilon}=u_{0}&\text{on}\,\,\mathbb{R}^{d}\times\\{0\\}.\end{array}\right.$

Here $m:\mathbb{T}^{d}\times S^{d-1}\to(0,\infty)$ is a mobility coefficient
and $A$ is obtained from a uniformly elliptic matrix field
$a:\mathbb{T}^{d}\times S^{d-1}\to\mathcal{S}_{d}$ through the following
relation:

(3) $A(y,e)=(\text{Id}-e\otimes e)a(y,e)(\text{Id}-e\otimes e).$

Though the mobility $m$ can be incorporated into $A$, we will see below that
it is illuminating to write the equation in this form.

We prove that when $m$ and $a$ are regular enough, there are effective
coefficients $\overline{m}$ and $\bar{a}$ such that, roughly speaking, if
$\bar{A}$ is related to $\bar{a}$ as in (3), then the solutions
$(u^{\epsilon})_{\epsilon>0}$ are described as $\epsilon\to 0^{+}$ by the
equation

(4)
$\overline{m}(\widehat{D\bar{u}})\bar{u}_{t}-\text{tr}\left(\bar{A}(\widehat{D\bar{u}})D^{2}\bar{u}\right)=0\quad\text{in}\,\,\mathbb{R}^{d}\times(0,\infty).$

However, in dimensions $d\geq 3$, we prove that the effective coefficients
$\bar{a}$ and $\bar{m}$ are generically discontinuous at normal directions
$e\in\mathbb{R}\mathbb{Z}^{d}$, so-called _rational directions_. We prove that
the solutions $(u^{\epsilon})_{\epsilon>0}$ are controlled by sub- and
supersolutions of (4) in general; it is not clear whether or not oscillations
persist in the limit.

Finally, we show that, in some cases, the effective mobility $\overline{m}$
describes the linear response of (2), precisely as in the work of Spohn [45]
and Katsoulakis and Souganidis [34], [35]. Again, there are pathologies in
rational directions that complicate the picture.

We expect that these results can be generalized to (1) with fully nonlinear
operators $F$ in arbitrary dimensions $d\geq 3$. Toward that end, it will be
necessary to overcome a number of difficulties that arise when the correctors
used in the asymptotic analysis are no longer $C^{2}$.

###  1.1. Literature review

The level-set method for describing geometric flows was introduced by Ohta,
Jasnow, and Kawasaki [40], Sethian [43], and Osher and Sethian [41]. The first
rigorous description of the method was developed by Barles [3] for a first
order model of flame propagation and, in the second order setting, by Evans
and Spruck [24] and Chen, Giga, and Goto [19]. See the papers of Barles,
Soner, and Souganidis [8] and Barles and Souganidis [9] for the state of the
art.

Much of the literature on the homogenization of second order level set PDE
treats the hyperbolic scaling, where the aim is to obtain a first order motion
in the limit. For periodic media, we refer the reader to the papers by Lions
and Souganidis [36], Caffarelli and Monneau [14], and Gao and Kim [30] and the
references therein. The random setting has been treated by Armstrong and
Cardaliaguet [1] and Feldman [26].

Fewer results are available in the parabolic scaling. Novaga and Valdinoci
[37] proved a pinning result for mean curvature with periodic forcing in this
scaling, showing that pinning is generic for forcing terms in the
$L^{1}(\mathbb{T}^{d})$ topology and relating this phenomenon to a prescribed
curvature problem. Barles, Cesaroni, and Novaga [4] prove homogenization of a
similar problem when the forcing is laminar and the initial datum is a graph
over a suitable direction.

A number of works have treated parabolicly scaled geometric motions subject to
small forcings in the same spirit as equation (20) below. Cesaroni, Novaga,
and Valdinoci [18] proved homogenization of plane-like solutions of a forced
mean curvature equation closely related to (20), obtaining homogenized speeds
that are discontinuous with respect to the direction. That built on the work
of Craciun and Bhattacharya [21], who had initially studied the homogenized
speeds in question, and is related to a paper by Chen and Lou [20], who found
planar and V-shaped solutions of equations like (2) and (20) in dimension two
and with almost periodic forcings.

During the preparation of this paper, the author learned that a number of the
ideas presented here were anticipated by Craciun and Bhattacharya in their
study of periodically forced mean curvature flow. Compare the proofs of
Theorem 8 and Proposition 17 below to the formal analysis of [21, Proposition
6].

Our approach draws a novel connection between the homogenization of second
order level set PDE and that of uniformly elliptic equations in nondivergence
form. Throughout the paper, we study the asymptotics of (1) using a degenerate
elliptic cell problem in the torus $\mathbb{T}^{d}$. As shown in Section 4
below, this cell problem leads to the analysis of a family of uniformly
elliptic equation with almost periodic coefficients, which falls under the
purview of the papers by Caffarelli, Souganidis, and Wang [16] and Caffarelli
and Souganidis [15]. In the study of the discontinuity properties of the
homogenized coefficients, in particular, we make extensive use of the obstacle
problem formulation of [16].

Finally, where the discontinuity of the effective coefficients is concerned,
our approach is inspired by recent developments in the study of oscillating
boundary value problems. A non-exhaustive list of references in this area
includes the papers by Barles, Da Lio, and Souganidis [5], Barles and
Mironescu [7], Gérard-Varet and Masmoudi [32], Choi and Kim [CK], Feldman
[25], Feldman and Kim [27], Feldman and Zhang [29], and Feldman, Kim, and
Souganidis [28]. This paper adapts to the setting of geometric flows the
approach of [27].

###  1.2. Organization of the Paper

In the next section, the main results of the paper are stated and the main
arguments are sketched. Section 4 is devoted to the analysis of the degenerate
cell problem used throughout the paper. In Section 5, these correctors are
used to show that the homogenized motion has the desired behavior whenever its
normal vector is irrational, and we explain how to use this to prove
homogenization of sub- and supersolutions. Section 6 treats the continuity
properties of the homogenized operator, including the proof that discontinuity
in rational directions is generic. Finally, Section 7 studies the effective
mobility of the quasilinear problem from the point of view of linear response.

A number of technical results needed in the paper are provided in the
appendix.

###  1.3. Notation

The Euclidean inner product in $\mathbb{R}^{d}$ is denoted by
$\langle\cdot,\cdot\rangle$; $\|\cdot\|$ is its associated norm. $S^{d-1}$ is
the unit sphere in $\mathbb{R}^{d}$, that is, those vectors $e$ with
$\|e\|=1$. We write $\langle e\rangle$ for the real span of $e$ and $\langle
e\rangle^{\perp}$ for its orthogonal complement.

Given $p\in\mathbb{R}^{d-1}\setminus\\{0\\}$, we write $\hat{p}=\|p\|^{-1}p$.

$\mathcal{S}_{d}$ is the space of real $d\times d$ symmetric matrices. Given
$e\in S^{d-1}$, $e\otimes e$ is the orthogonal projection onto $\langle
e\rangle$. Given such an $e$ and an $X\in\mathcal{S}_{d}$, we define the
matrix $\tilde{X}_{e}$ by $\tilde{X}_{e}=(\text{Id}-e\otimes
e)X(\text{Id}-e\otimes e)$.

$\mathbb{T}^{d}$ is the $d$-dimensional torus, that is, the quotient space
obtained from $\mathbb{R}^{d}$ by identifying points that differ by an element
of the integers $\mathbb{Z}^{d}$. $\mathbb{R}\mathbb{Z}^{d}$ denotes the real
span of $\mathbb{Z}^{d}$.

Given an $e\in S^{d-1}$, the differential operator $D^{2}_{e}$ is defined via
its action on smooth functions $\varphi$ by
$D^{2}_{e}\varphi=(\text{Id}-e\otimes e)D^{2}\varphi(\text{Id}-e\otimes e)$.

$\mathcal{L}^{d}$ and $\mathcal{H}^{d-1}$ are the Lebesgue measure and
$(d-1)$-dimensional Hausdorff measure in $\mathbb{T}^{d}$, respectively, the
latter normalized to coincide with surface area.

##  2\. Main Results

###  2.1. Homogenization for $d=2$

In dimension two, we consider operators
$F:(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}\times\mathbb{T}^{2}\to\mathbb{R}$
satisfying the following assumptions:

  * (i)

(Geometric) If $\nu,\mu\in\mathbb{R}$ and $\nu>0$, then

(5) $F(\nu p,\nu X+\mu p\otimes p,y)=\nu
F(p,X,y)\quad\text{for}\,\,(p,X,y)\in\mathbb{R}^{2}\times\mathcal{S}_{2}\times\mathbb{T}^{2}.$

  * (ii)

(Stationary planes) For each $e\in S^{1}$,

(6) $F(e,0,y)=0\quad\text{for}\,\,y\in\mathbb{T}^{2}.$

  * (iii)

(Uniform degenerate ellipticity) There are constants $\lambda,\Lambda>0$ such
that if $(e,X,y)\in S^{1}\times\mathcal{S}_{2}\times\mathbb{T}^{2}$,
$Y\in\mathcal{S}_{2}$ satisfies $Y\geq 0$, and $\tilde{Y}_{e}$ is the matrix
$\tilde{Y}_{e}=(\text{Id}-e\otimes e)Y(\text{Id}-e\otimes e)$, then

(7) $\lambda\|\tilde{Y}_{e}\|\leq
F(e,X+Y,y)-F(e,X,y)\leq\Lambda\|\tilde{Y}_{e}\|.$

  * (iv)

(Regularity) $F$ is continuous in
$(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}\times\mathbb{T}^{2}$.

  * (v)

(Comparison) $F$ satisfies a technical assumption ensuring (1) is well-posed.
(This assumption is stated precisely in the appendix.)

######  Theorem 1.

If $F$ satisfies assumptions (i)-(v) above, then there is a continuous
function
$\overline{F}:(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}\to\mathbb{R}$
satisfying the same assumptions such that if $u_{0}\in UC(\mathbb{R}^{2})$,
$(u^{\epsilon})_{\epsilon>0}$ are the unique viscosity solutions of (1), and
$\bar{u}:\mathbb{R}^{d}\times[0,\infty)\to\mathbb{R}$ is the solution of the
effective equation

$\left\\{\begin{array}[]{r
l}\bar{u}_{t}-\overline{F}\left(D\bar{u},D^{2}\bar{u}\right)=0&\text{in}\,\,\mathbb{R}^{2}\times(0,\infty),\\\
\bar{u}=u_{0}&\text{on}\,\,\mathbb{R}^{2}\times\\{0\\},\end{array}\right.$

then $u^{\epsilon}\to\bar{u}$ locally uniformly in
$\mathbb{R}^{2}\times[0,\infty)$.

Well-posedness of (1) is reviewed in Appendix A.1 below.

###  2.2. Quasi-linear Operators with Regular Coefficients

We now state the results concerning the homogenization of (2). Here and
henceforth we fix $\lambda,\Lambda>0$ and define the space
$\mathcal{S}_{d}(\lambda,\Lambda)$ by

$\mathcal{S}_{d}(\lambda,\Lambda)=\left\\{a_{0}\in\mathcal{S}_{d}\,\mid\,\lambda\text{Id}<a_{0}<\Lambda\text{Id}\right\\}.$

The matrix field $a:\mathbb{T}^{d}\times S^{d-1}\to\mathcal{S}_{d}$ is assumed
to satisfy the following:

(8) $\displaystyle a$ $\displaystyle:\mathbb{T}^{d}\times
S^{d-1}\to\mathcal{S}_{d}(\lambda,\Lambda)\quad\text{is continuous},$ (9)
$\displaystyle\|a(\cdot,e)\|_{C^{2,\alpha}(\mathbb{T}^{d})}<\infty\quad\text{if}\,\,e\in
S^{d-1},$ (10) $\displaystyle\sup$
$\displaystyle\left\\{\frac{\|a(y,e)-a(y^{\prime},e)\|}{\|y-y^{\prime}\|}\,\mid\,(e,y),(e,y^{\prime})\in
S^{d-1}\times\mathbb{T}^{d},\,\,y^{\prime}\neq y\right\\}<\infty.$

Concerning the mobility $m:\mathbb{T}^{d}\times S^{d-1}\to(0,\infty)$, the
assumptions are listed next:

(11) $\displaystyle m\,\,$ $\displaystyle\text{is continuous},$ (12)
$\displaystyle\|m(\cdot,e)$
$\displaystyle\|_{C^{2,\alpha}(\mathbb{T}^{d})}<\infty\quad\text{if}\,\,e\in
S^{d-1},$ (13) $\displaystyle\sup\bigg{\\{}$
$\displaystyle\frac{|m(y,e)-m(y^{\prime},e)|}{\|y-y^{\prime}\|}\,\mid\,(y,e),(y^{\prime},e)\in\mathbb{T}^{d}\times
S^{d-1},\,\,y\neq y^{\prime}\bigg{\\}}<\infty.$

The main result in this setting is:

######  Theorem 2.

If $a$ satisfies assumptions (8), (9), and (10) and $m$ satisfies (11), (12),
and (13), then there are effective coefficients
$\overline{m}:S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}\to(0,\infty)$ and
$\bar{a}:S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}\to\mathcal{S}_{d}$ such that
if
$\overline{F}:(\mathbb{R}^{d}\setminus\mathbb{R}\mathbb{Z}^{d})\times\mathcal{S}_{d}\to\mathbb{R}$
is given by

(14)
$\overline{F}(p,X)=\overline{m}(\hat{p})^{-1}\text{tr}\left(\bar{a}(\hat{p})\tilde{X}_{\hat{p}}\right),$

$u_{0}\in UC(\mathbb{R}^{d})$, and $(u^{\epsilon})_{\epsilon>0}$ are the
solutions of (2), then the half-relaxed limits $\bar{u}^{*}$ and $\bar{u}_{*}$
given by

$\bar{u}^{*}=\limsup\nolimits^{*}u^{\epsilon},\quad\bar{u}_{*}=\liminf\nolimits_{*}u^{\epsilon},$

are sub- and supersolution, respectively, of the following problems:

$\left\\{\begin{array}[]{r
l}\bar{u}^{*}_{t}-\overline{F}^{*}(D\bar{u}^{*},D^{2}\bar{u}^{*})\leq
0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty)\\\
\bar{u}^{*}=u_{0}&\text{in}\,\,\mathbb{R}^{d}\times\\{0\\}\end{array}\right.\left\\{\begin{array}[]{r
l}\bar{u}_{*,t}-\overline{F}_{*}(D\bar{u}_{*},D^{2}\bar{u}_{*})\geq
0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty)\\\
\bar{u}_{*}=u_{0}&\text{on}\,\,\mathbb{R}^{d}\times\\{0\\}\end{array}\right.$

Well-posedness of (2) is reviewed in Appendix A.1.

Even though we have imposed considerable regularity restrictions on the matrix
field $a$, nonetheless we show below that $\overline{F}$ generically fails to
admit a continuous extension. We expect this is true more generally (i.e.
fully non-linear operators) whenever $d\geq 3$.

In certain special cases, $\overline{F}^{*}=\overline{F}_{*}$ in Theorem 2
and, thus, homogenization occurs as in Theorem 1. Some examples of this are
discussed in Section 3.1 below.

######  Remark 1.

The smoothness assumptions on $m$ and $a$ are certainly restrictive. As we
will see below, these assumptions are natural in light of the degeneracy of
the cell problems used in the asymptotic analysis. At various places in the
arguments, it is not clear how to proceed unless the corrector is $C^{2}$, and
requiring the coefficients to be $C^{2}$ is a natural way to achieve this.

Nonetheless, when $d=2$, we are able to obtain $C^{2}$ approximate correctors
by regularizing the coefficients and exploiting the special structure of the
cell problem in this dimension. It is for this reason that Theorem 1 is more
general.

We expect that, using the same techniques, Theorem 2 could be extended to
convex or concave operators that are twice continuously differentiable in the
$(X,y)$ variables, but this is not so interesting since the main examples we
have in mind are one-homogeneous in the Hessian.

###  2.3. (Dis)continuity of the Homogenized Coefficients

The effective operator $\overline{F}$ in Theorem 2 is determined by the
invariant probability measures associated with a certain family of diffusion
processes on the torus. Toward that end, we define the sets of invariant
measures $\\{\mathscr{I}^{a}_{e}\\}_{e\in
S^{d-1}}\subseteq\mathscr{P}(\mathbb{T}^{d})$ so that
$\mu\in\mathcal{I}_{e}^{a}$ if and only if, for each $\varphi\in
C^{\infty}(\mathbb{T}^{d})$, writing $D^{2}_{e}\varphi=(\text{Id}-e\otimes
e)D^{2}\varphi(\text{Id}-e\otimes e)$, we have

$\int_{\mathbb{T}^{d}}\text{tr}\left(a(y,e)D^{2}_{e}\varphi(y)\right)\mu(dy)=0.$

For a given $e\in S^{d-1}$, these are precisely the invariant probability
measures of the diffusion process $X^{e}$ on $\mathbb{T}^{d}$ determined by
the SDE

(15) $dX^{e}_{t}=(\text{Id}-e\otimes
e)\sqrt{a(X^{e}_{t},e)}(\text{Id}-e\otimes e)dB_{t}.$

The basic structure of the sets $\\{\mathcal{I}_{e}^{a}\\}_{e\in S^{d-1}}$ and
their relationship to the homogenized coefficients $\bar{a}$ and
$\overline{m}$ is summed up in the following theorem. In the rational case,
the foliation of $\mathbb{T}^{d}$ into sub-tori normal to $e$ arises
naturally. That is, we will use the decomposition of $\mathbb{T}^{d}$ as

$\mathbb{T}^{d}=\bigcup_{r\in[0,r_{e})}\mathbb{T}^{d-1}_{e}(r),$

where the sub-tori $\\{\mathbb{T}^{d-1}_{e}(r)\\}_{r\in[0,r_{e})}$ and the
period $r_{e}>0$ are defined by

(16) $\displaystyle\mathbb{T}^{d-1}_{e}(r)$
$\displaystyle=\left\\{y\in\mathbb{T}^{d}\,\mid\,\langle y,e\rangle=r+\langle
k,e\rangle\,\,\text{for some}\,\,k\in\mathbb{Z}^{d}\right\\},$ $\displaystyle
r_{e}$ $\displaystyle=\min\left\\{\langle
k,e\rangle\,\mid\,k\in\mathbb{Z}^{d}\right\\}\cap(0,\infty).$

When $e$ is rational, the trajectories of the process $X^{e}$ of (15) remain
confined to $\mathbb{T}^{d-1}_{e}(\langle X^{e}_{0},e\rangle)$ and this leads
to a corresponding decomposition of $\mathscr{I}^{a}_{e}$.

Now we are prepared to state the main result concerning the structure of the
invariant measures and their relation to the effective coefficients $\bar{a}$
and $\overline{m}$:

######  Theorem 3.

Under the assumptions of Theorem 2, we have:

  * (i)

If $e\notin\mathbb{R}\mathbb{Z}^{d}$, then there is a unique probability
measure $\bar{\mu}_{e}$ such that $\mathscr{I}^{a}_{e}=\\{\bar{\mu}_{e}\\}$.
Furthermore, $\bar{\mu}_{e}\ll\mathcal{L}^{d}$.

  * (ii)

If $e\in\mathbb{R}\mathbb{Z}^{d}$, then there is an $r_{e}$-periodic function
$\mu_{e}:\mathbb{R}\to\mathscr{P}(\mathbb{T}^{d})$,
$\mu_{e}:s\mapsto\mu_{e}^{s}$, such that $\mathscr{I}^{a}_{e}$ equals the
closed convex hull of $\\{\mu_{e}^{s}\,\mid\,s\in\mathbb{R}\\}$. For each
$s\in[0,r_{e})$, we have
$\mu_{e}^{s}\ll\mathcal{H}^{d-1}\restriction_{\mathbb{T}^{d-1}_{e}(r)}$.

  * (iii)

For each $e\in S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$, $\bar{a}$ and
$\overline{m}$ are given by

(17)
$\overline{a}(e)=\int_{\mathbb{T}^{d}}a(y,e)\,\bar{\mu}_{e}(dy),\quad\overline{m}(e)=\int_{\mathbb{T}^{d}}m(y,e)\,\bar{\mu}_{e}(dy).$

Theorems 2 and 3 lead to the question of the continuity properties of the set-
valued maps $e\mapsto\mathscr{I}^{a}_{e}$. When
$e\notin\mathbb{R}\mathbb{Z}^{d}$, continuity at $e$ is implied directly by
uniqueness and compactness. The continuity question is significantly more
complicated when $e\in\mathbb{R}\mathbb{Z}^{d}$. The next results address that
issue.

First, we need a digression on analysis in $S^{d-1}$ because it turns out that
the limiting behavior of $e\mapsto\mathscr{I}^{a}_{e}$ at a rational direction
is dependant on the direction of approach. To make that precise, notice that
if $(e_{n})_{n\in\mathbb{N}}\subseteq S^{d-1}$ converges to $e$ as
$n\to\infty$, then there is necessarily a a subsequence
$(n_{j})_{j\in\mathbb{N}}\subseteq\mathbb{N}$ and an $\eta\in
S^{d-1}\cap\langle e\rangle^{\perp}$ such that
$-\eta=\lim_{j\to\infty}\frac{e_{n_{j}}-e}{\|e_{n_{j}}-e\|}$. Geometrically,
that means $e_{n_{j}}$ approximately approaches $e$ along the great circle
parametrized by $\theta\mapsto\cos(\theta)e+\sin(\theta)\eta$, or, put simply,
$e_{n_{j}}\to e$ along the $\eta$ direction. We will see that it is necessary
to take account of the direction $\eta$ when studying the continuity
properties of $e\mapsto\mathscr{I}^{a}_{e}$.

Finally, in the statement, it is convenient to use the metrizability of
$\mathscr{P}(\mathbb{T}^{d})$. Toward that end, we fix here and henceforth a
metric
$D:\mathscr{P}(\mathbb{T}^{d})\times\mathscr{P}(\mathbb{T}^{d})\to[0,\infty)$
inducing the weak-$*$ topology (e.g. Wasserstein distance).

######  Theorem 4.

Under the assumptions of Theorem 2, for each $e\in\mathbb{R}\mathbb{Z}^{d}$,
if we define the function $\tilde{\mu}_{e}:S^{d-1}\cap\langle
e\rangle^{\perp}\to\mathscr{P}(\mathbb{T}^{d})$,
$\tilde{\mu}_{e}:\eta\to\tilde{\mu}_{e}^{\eta}$, by

(18) $\displaystyle\tilde{\mu}_{e}^{\eta}$
$\displaystyle=\left(\int_{0}^{r_{e}}\langle
a^{\perp}_{e}(se)\eta,\eta\rangle^{-1}\,ds\right)^{-1}\int_{0}^{r_{e}}\langle
a^{\perp}_{e}(se)\eta,\eta\rangle^{-1}\mu^{s}_{e}\,ds,$ (19) $\displaystyle
a_{e}^{\perp}(y)$
$\displaystyle=\int_{\mathbb{T}^{d}}a(y^{\prime},e)\mu_{e}^{\langle
y,e\rangle}(dy^{\prime}),$

then this function describes the continuity properties of
$e^{\prime}\mapsto\mathscr{I}^{a}_{e^{\prime}}$ at $e$ in the sense that

$\lim_{\delta\to
0^{+}}\sup\left\\{D(\mu,\tilde{\mu}_{e}^{\eta})\,\mid\,\mu\in\mathscr{I}^{a}_{e^{\prime}},\,\,e^{\prime}\notin\mathbb{R}\mathbb{Z}^{d},\,\,\|e^{\prime}-e\|+\left\|\frac{e^{\prime}-e}{\|e^{\prime}-e\|}+\eta\right\|<\delta\right\\}=0.$

The corollary that follows shows that discontinuity is generic among
coefficients $a$ satisfying the assumptions of Theorem 2 if $d\geq 3$.

######  Corollary 1.

Assume $d\geq 3$. There is a residual set $\mathscr{C}_{d}\subseteq
C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))$ such that if
$a$ is independent of the $e$ variable and $a\in\mathscr{C}_{d}$, then the
following statements hold:

  * (a)

If $\tilde{\mu}$ is the function defined in Theorem 4, then

$\forall e\in
S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}\quad\\#\\{\tilde{\mu}_{e}^{\eta}\,\mid\,\eta\in
S^{d-1}\cap\langle e\rangle^{\perp}\\}=\infty.$

  * (b)

The effective coefficient
$\bar{a}:S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}\to\mathcal{S}_{d}(\lambda,\Lambda)$
defined in Theorem 3 has infinitely many distinct directional limits at each
$e\in\mathbb{R}\mathbb{Z}^{d}$.

In particular, by taking $a\in\mathscr{C}_{d}$ and $m\equiv 1$ in Theorem 2,
we find that $\overline{F}^{*}\neq\overline{F}_{*}$.

###  2.4. Effective Mobility and Linear Response

Finally, we discuss the relationship between the effective mobility
$\overline{m}$ in (2) and linear response. Specifically, given $e\in S^{d-1}$
and defining $A$ as in (3), we are interested in the perturbed problem

(20) $\left\\{\begin{array}[]{r
l}m(\epsilon^{-1}x,\widehat{Du_{e}^{\epsilon}})u^{\epsilon}_{e,t}-\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du_{e}^{\epsilon}})D^{2}u_{e}^{\epsilon}\right)-\alpha\|Du_{e}^{\epsilon}\|=0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty),\\\
u_{e}^{\epsilon}(x,0)=\langle
x,e\rangle&\text{if}\,\,x\in\mathbb{R}^{d}.\end{array}\right.$

To start with, we prove homogenization of (20):

######  Theorem 5.

Fix $d\geq 2$. If $a$ and $m$ satisfy the assumptions of Theorem 2, then there
is an $\overline{m}_{\text{pl}}:S^{d-1}\to(0,\infty)$ such that if $e\in
S^{d-1}$ and $(u_{e}^{\epsilon})_{\epsilon>0}$ are the solutions of (20), then

$\lim_{\epsilon\to 0^{+}}u_{e}^{\epsilon}(x,t)=\langle
x,e\rangle+\alpha\overline{m}_{\text{pl}}(e)^{-1}t\quad\text{locally uniformly
in}\,\,\mathbb{R}^{d}\times[0,\infty).$

When $d=2$, $\overline{m}_{\text{pl}}^{\,-1}$ is precisely the derivative of
the front speeds obtained by Caffarelli and Monneau [14]. Recall that if $e\in
S^{1}$ and $\alpha\in\mathbb{R}$ and if $u^{e,\alpha}$ is the solution of the
forced problem

(21) $\left\\{\begin{array}[]{r
l}m(y,\widehat{Du^{e,\alpha}})u^{e,\alpha}_{t}-\text{tr}\left(A(y,\widehat{Du^{e,\alpha}})D^{2}u^{e,\alpha}\right)-\alpha\|Du^{e,\alpha}\|=0&\text{in}\,\,\mathbb{R}^{2}\times(0,\infty),\\\
u^{e,\alpha}(x)=\langle
x,e\rangle&\text{if}\,\,x\in\mathbb{R}^{2},\end{array}\right.$

then, by [14], there is a $\lambda_{e}(\alpha)\in\mathbb{R}$ such that

$\lim_{R\to\infty}R^{-1}u^{e,\alpha}(Rx,Rt)=\langle
x,e\rangle+\lambda_{e}(\alpha)t\quad\text{locally uniformly
in}\,\,\mathbb{R}^{2}\times[0,\infty).$

The derivative of this function is $\overline{m}_{\text{pl}}^{\,-1}$:

######  Corollary 2.

If $d=2$ and $e\in S^{1}$, then
$\overline{m}_{\text{pl}}(e)^{-1}=\lim_{\alpha\to
0}\alpha^{-1}\lambda_{e}(\alpha)$.

Notice that, up to a parabolic rescaling, the solution $u^{\epsilon}_{e}$ of
(20) equals $u^{e,\alpha(\epsilon)}$ with $\alpha(\epsilon)=\alpha\epsilon$.
As discussed by Spohn [45] and Bellettini, Buttà, and Presutti [10], this is
precisely the scaling where the forcing and the curvature are evenly matched,
and the limiting behavior of $u^{\epsilon}$ should capture the linear response
of (2) to small perturbations. In light of previous works, it is natural to
expect that $\overline{m}_{\text{pl}}=\overline{m}$.

It turns out these functions only coincide in irrational directions, even when
$d=2$. To start with, we obtain an expression for $\overline{m}_{\text{pl}}$:

######  Theorem 6.

Assume that $d\geq 2$ and $a$ and $m$ satisfy the assumptions of Theorem 2.
The following statements hold:

  * (i)

If $e\in S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$, then
$\overline{m}_{\text{pl}}(e)=\overline{m}(e)$.

  * (ii)

If $e\in S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$ and $\mu_{e}$ is the function
defined in Theorem 3, then

(22)
$\overline{m}_{\text{pl}}(e)=r_{e}^{-1}\int_{0}^{r_{e}}\int_{\mathbb{T}^{d}}m(y,e)\,\mu^{s}_{e}(dy)\,ds.$

In view of the formula for $\overline{m}_{\text{pl}}$ in rational directions,
a proof similar to that of Corollary 1 yields the following:

######  Corollary 3.

If $d\geq 2$, then there is a residual set $\mathcal{A}_{d}\subseteq
C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))\times
C^{2,\alpha}(\mathbb{T}^{d};(0,\infty))$ such that if $m$ and $a$ are
independent of $e$ and $(a,m)\in\mathcal{A}_{d}$, then, for each $e\in
S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$, the following hold:

  * (i)

There is an $\eta\in S^{d-1}\cap\langle e\rangle^{\perp}$ such that
$\overline{m}_{\text{pl}}(e)\neq\tilde{m}_{e}^{\eta}$.

  * (ii)

If $d\geq 3$, then
$\overline{m}:S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}\to(0,\infty)$ does not
have a limit at $e$.

In particular, if $(a,m)\in\mathcal{A}_{d}$, then $\overline{m}_{\text{pl}}$
is not a continuous function.

###  2.5. Strategy of Proof

Let us briefly review the main elements of the proof of Theorems 1, 2, and 5.
Inspired by the approach of Barles and Souganidis [9] and Barles, Cesaroni,
and Novaga [4], we employ the ansatz

$u^{\epsilon}(x,t)=\bar{u}(x,t)+\epsilon^{2}V(\epsilon^{-1}x)+\dots$

which leads to the following cell problems for correctors $V$ depending on the
normal vector $e\in S^{d-1}$ and second fundamental form
$X\in\mathcal{S}_{d}$:

(23) $-F(e,X+D^{2}V,y)=-\overline{F}(e,X)\quad\text{if}\,\,\mathbb{T}^{d}.$

In view of (7), this is a degenerate elliptic PDE. The analysis below shows
that it is ill-posed when $e\in\mathbb{R}\mathbb{Z}^{d}$, which poses a
significant obstacle in what follows.

Nonetheless, in _irrational directions_ , that is, if
$e\notin\mathbb{R}\mathbb{Z}^{d}$, it turns out that if we let
$(V^{\delta})_{\delta>0}$ be the solutions of the penalized cell problem

(24) $\delta V^{\delta}-F(e,X+D^{2}V,y)=0\quad\text{in}\,\,\mathbb{T}^{d},$

then there is an $\overline{F}(e,X)\in\mathbb{R}$ such that

(25) $\overline{F}(e,X)=\lim_{\delta\to 0^{+}}\delta
V^{\delta}\quad\text{uniformly in}\,\,\mathbb{T}^{d}.$

As we show below, this can be seen by rewriting (24) as a one-parameter family
of uniformly elliptic $(d-1)$-dimensional problems and applying homogenization
results for almost periodic, uniformly elliptic operators in nondivergence
form.

Once (25) is proved, we construct approximate correctors of (24) following a
well-worn approach in homogenization theory. The approximate correctors are
then used to show that the solutions $(u^{\epsilon})_{\epsilon>0}$ behave
appropriately wherever the limiting level set has an irrational normal.

That leaves the question of whether or not a function that solves a level set
PDE at points where its normal vector is irrational is actually a genuine
solution. In this case, as long as we are studying the unforced problems (1)
or (2), the answer is yes and follows from the results of the companion paper
[39], as shown below.

Two problems remain. First, it is not at all clear that the function
$\overline{F}$ extends to a continuous function. Toward that end, we adapt
ideas from the study of oscillating boundary value problems to characterize
all possible limits of $\overline{F}$ at rational directions.

The remaining issue is the homogenization of the forced problem (20). Here
additional ideas are needed to treat rational directions. If
$e\in\mathbb{R}\mathbb{Z}^{d}$ and we let $(V^{\delta})_{\delta>0}$ be the
solutions of

(26) $m(y,e)+\delta
V^{\delta}-\text{tr}\left(A(y,e)D^{2}V^{\delta}\right)=0\quad\text{in}\,\,\mathbb{T}^{d},$

then there is an $m^{\perp}_{e}\in C(\mathbb{T}^{d})$ varying only in the $e$
direction such that

$m^{\perp}_{e}=\lim_{\delta\to 0^{+}}(-\delta V^{\delta})\quad\text{uniformly
in}\,\,\mathbb{T}^{d}.$

In general, $m^{\perp}_{e}$ is not a constant function. This reason is that,
at the level of (20), sending $\delta\to 0^{+}$ in (26) only averages the
fluctuations of the front relative to a moving reference plane. As we show
below, the height of this plane oscillates around its mean and needs to be
corrected through another, one-dimensional cell problem.

##  3\. Examples

In this section, we discuss a few examples of interest that fit into the
assumptions of either Theorem 1 or Theorem 2.

###  3.1. Anisotropic Curvature Flows with Periodic Mobilities

Let $\varphi:\mathbb{R}^{d}\to[0,\infty)$ be a $C^{2}$ Finsler norm, that is,
a convex, positively one-homogeneous function that is $C^{2}$ in
$\mathbb{R}^{d}\setminus\\{0\\}$. Given a mobility coefficient $m$ as above,
the equation

(27)
$m(y,\widehat{Du})u_{t}-\text{tr}\left(D^{2}\varphi(\widehat{Du})D^{2}u\right)=0\quad\text{in}\,\,\mathbb{R}^{d}\times(0,\infty)$

is referred to as an anisotropic curvature flow. Formally, this is the
gradient flow of the anisotropic perimeter determined by $\varphi$ with
respect to the $L^{2}$-Riemannian metric determined by $m$ (see [46]). The
geometric flows associated with these equations are of interest in materials
science.

Since here $a(y,e)=D^{2}\varphi(e)$ is independent of $y$, the results of
Theorems 2 and 4 imply that homogenization occurs. We state the result,
leaving the details to the reader:

######  Proposition 1.

If $\varphi$ is a $C^{2}$ Finsler norm, $m$ satisfies the assumptions of
Theorem 2, and $a(y,e)=D^{2}\varphi(e)\in\mathcal{S}_{d}(\lambda,\Lambda)$ for
all $e\in S^{d-1}$, then the solutions $(u^{\epsilon})_{\epsilon>0}$ of (2)
converge locally uniformly to the solution $\bar{u}$ of the averaged equation

$\left\\{\begin{array}[]{r
l}\overline{m}(\widehat{D\bar{u}})\bar{u}_{t}-\text{tr}\left(D^{2}\varphi(\widehat{D\bar{u}})D^{2}\bar{u}\right)=0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty),\\\
\bar{u}=u_{0}&\text{on}\,\,\mathbb{R}^{d}\times\\{0\\}.\end{array}\right.$

Here $\overline{m}(e)=\int_{\mathbb{T}^{d}}m(y,e)\,dy$. Furthermore, in this
case, $\overline{m}_{\text{pl}}=\overline{m}$ in $S^{d-1}$.

It is interesting to note that $\overline{m}_{\text{pl}}$ is continuous in
this set-up. By contrast, given a positive function $c\in
C^{0,1}(\mathbb{T}^{d})$, the structurally similar forced problem

$\left\\{\begin{array}[]{r
l}u^{\epsilon}_{t}-\text{tr}\left(D^{2}\varphi(\widehat{Du^{\epsilon}})D^{2}u^{\epsilon}\right)+c(\epsilon^{-1}x)\|Du^{\epsilon}\|=0&\text{in}\,\,\mathbb{R}^{d}\times(0,\infty),\\\
u^{\epsilon}(x,0)=\langle
x,e\rangle&\text{on}\,\,\mathbb{R}^{d}\times(0,\infty),\end{array}\right.$

also homogenizes with $u^{\epsilon}(x,t)\to\langle x,e\rangle+\bar{c}(e)t$ in
the limit. (This was proved when $d=2$ and $\varphi=\|\cdot\|$ by [18] and
follows by arguing as in Section 7 below in general.) However, as in [18], one
finds that $\bar{c}$ is generically discontinuous. Put slightly differently,
pathologies arise if we replace constant forcing by periodic forcing in (20).

Let us observe that by exploiting the regularity theory of the Laplacian and
using approximation arguments as in [39], it is not hard to show that
Proposition 1 still holds if (12) is dropped.

Finally, a few remarks are in order:

######  Remark 2.

Of course, it would also be interesting to analyze the case when $\varphi$
depends on $y$ as well as $e$. In that case, to get an anisotropic curvature
flow, (27) should include another term involving $D\varphi$ and no longer fits
into the framework of (2). The homogenization of such equations, which are the
natural divergence form analogue of (2), is largely open. However, see [4] for
results that apply to the case where $\varphi(y,e)=1+\langle\psi(y),e\rangle$
for some vector field $\psi$ satisfying
$\|\psi\|_{L^{\infty}(\mathbb{T}^{d})}<1$.

######  Remark 3.

In general, one obtains $\overline{F}^{*}=\overline{F}_{*}$, and, thus,
homogenization, in Theorem 2 when $a$ is independent of the spatial variable,
as in the last proposition. Another (actually equivalent) case is when
$a(y,e)=\tilde{a}(y,e)\text{Id}$ for some positive function $\tilde{a}\in
C^{0,1}(\mathbb{T}^{d})$. In this case, $\bar{a}(e)$ equals the harmonic mean
of $\tilde{a}(\cdot,e)$, consistent with [20, Theorem C].

###  3.2. Fully Nonlinear Operators in Dimension Two

Notice that if $F$ satisfies assumptions (i)-(iv) of Theorem 1 and also is
Lipschitz continuous in the following sense

$|F(p,X,y)-F(p,X,y^{\prime})|\leq
C\|\tilde{X}_{e}\|\|y-y^{\prime}\|\quad\text{if}\,\,(p,X)\in(\mathbb{R}^{d}\setminus\\{0\\})\times\mathcal{S}_{d},\,\,y,y^{\prime}\in\mathbb{T}^{d},$

then we can argue as in [33] to see that $F$ also satisfies (v).

A natural class of fully nonlinear operators satisfying these assumptions take
the following form:

$F(p,X,y)=\sup_{\alpha\in\mathcal{A}}\inf_{\beta\in\mathscr{B}}\text{tr}\left(\left(\text{Id}-\hat{p}\otimes\hat{p}\right)a_{\alpha,\beta}(y,\hat{p})\left(\text{Id}-e\otimes
e\right)X\right).$

Here we assume that the matrix fields
$\\{a_{\alpha,\beta}\,\mid\,\alpha\in\mathcal{A},\,\,\beta\in\mathscr{B}\\}$
map $\mathbb{T}^{d}\times S^{d-1}$ continuously into
$\mathcal{S}_{d}(\lambda,\Lambda)$ and that there is an $L>0$ such that, for
each $(\alpha,\beta)\in\mathcal{A}\times\mathscr{B}$, we have

$\sup\left\\{\frac{\|a_{\alpha,\beta}(y,e)-a_{\alpha,\beta}(y^{\prime},e)\|}{\|y-y^{\prime}\|}\,\mid\,(y,e),(y^{\prime},e)\in\mathbb{T}^{d}\times
S^{d-1},\,\,y\neq y^{\prime}\right\\}\leq L.$

When $\mathcal{A}$ is a singleton so that $F$ is concave, such operators
describe stochastic target problems, which are of interest in financial
mathematics (see [44], [12]).

##  4\. Approximate Correctors

In this section, we construct approximate correctors that will be used in the
analysis of the asymptotics of (1) and (2). The use of approximate correctors
in homogenization is by now standard (cf. [36] and [16]). The main difficulty
here is the operators of interest are degenerate elliptic and, therefore, as
we will see below, the cell problem is not well behaved in rational
directions.

The result we need concerning approximate correctors is stated next. The
majority of this section is devoted to its proof.

######  Theorem 7.

If $F$ satisfies the assumptions of Theorem 1 or $F$ takes the form

$F(p,X,y)=m(y,\hat{p})^{-1}\text{tr}(a(y,\hat{p})\tilde{X}_{\hat{p}})$

with $m$ and $a$ satisfying the assumptions of Theorem 2, then, for each
$(e,X)\in S^{d-1}\times\mathcal{S}_{d}$, there is a unique continuous function
$F^{\perp}_{e}(X,\cdot):\mathbb{T}^{d}\to\mathbb{R}$ varying only in the $e$
direction such that, for each $\nu>0$, there is a (non-unique) $V^{\nu}\in
C^{2}(\mathbb{T}^{d})$ satisfying

(28) $-\nu\leq F^{\perp}_{e}(X,\langle y,e\rangle
e)-F(e,\tilde{X}_{e}+D^{2}_{e}V^{\nu},y)\leq\nu\quad\text{in}\,\,\mathbb{T}^{d}.$

If $e\in S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$, then there is a constant
$\overline{F}(e,X)$ such that $F_{e}^{\perp}(X,\cdot)\equiv\overline{F}(e,X)$
in $\mathbb{T}^{d}$. Furthermore, $\overline{F}(e,\cdot)$ is continuous, it
satisfies (7), $|\overline{F}(e,X)|\leq\Lambda\|X\|$, and

$\overline{F}(e,X)=\overline{F}(e,\tilde{X}_{e})\quad\text{if}\,\,X\in\mathcal{S}_{d}.$

Following the proof of Theorem 7, we describe the relation between the
functions $\\{F_{e}^{\perp}\\}_{e\in S^{d-1}}$ and $\overline{F}$ and the
invariant measures of Theorem 3. Finally, at the end of the section, we prove
results specific to the rational case that are used in the proofs of Theorem 5
and Corollary 1.

In Appendix A.4, we discuss sufficient conditions for the existence of a
corrector (i.e. a solution of (28) with $\nu=0$) in the case when $a$ is
independent of the spatial variable.

In this section and elsewhere in the paper, we will extensively use the
differential operator $D^{2}_{e}$ defined by its action on smooth functions
$\varphi$ in $\mathbb{R}^{d}$ by

(29) $D^{2}_{e}\varphi=(\text{Id}-e\otimes e)D^{2}\varphi(\text{Id}-e\otimes
e).$

###  4.1. Cell Problem and Ergodic Constant

To start with, we establish the existence of penalized correctors and discuss
their convergence as $\delta\to 0^{+}$.

######  Theorem 8.

Given $e\in S^{d-1}$, $X\in\mathcal{S}_{d}$, and $\delta>0$, there is a unique
$V^{\delta}\in C(\mathbb{T}^{d})$ solving the penalized cell problem

(30) $\delta
V^{\delta}-F(e,X+D^{2}V^{\delta},y)=0\quad\text{in}\,\,\mathbb{T}^{d}.$

Furthermore, there is a function
$F^{\perp}_{e}(X,\cdot):\mathbb{T}^{d}\to\mathbb{R}$ varying only in the $e$
direction such that

(31) $\lim_{\delta\to 0^{+}}\sup\left\\{|\delta
V^{\delta}(y)+F^{\perp}_{e}(X,y)|\,\mid\,y\in\mathbb{T}^{d}\right\\}=0.$

In particular, if $e\notin\mathbb{R}\mathbb{Z}^{d}$, then there is a constant
$\overline{F}(e,X)\in\mathbb{R}$ such that
$F^{\perp}_{e}(X,\cdot)\equiv\overline{F}(e,X)$ in $\mathbb{T}^{d}$.

###### Proof.

The existence and uniqueness of $V^{\delta}\in C(\mathbb{T}^{d})$ is proved in
Appendix A.

Given $y\in\mathbb{T}^{d}$, define $\tilde{V}^{\delta}_{y}:\langle
e\rangle^{\perp}\to\mathbb{R}$ by

(32) $\tilde{V}^{\delta}_{y}(x^{\prime})=V^{\delta}(y+x^{\prime}).$

A perturbation argument (cf. [38, Appendix B]) shows that, no matter the
choice of $y$, the function $\tilde{V}^{\delta}_{y}$ is solution of

(33)
$\delta\tilde{V}^{\delta}_{y}-F(e,\tilde{X}_{e}+D^{2}_{e}\tilde{V}^{\delta}_{y},y+x^{\prime})=0\quad\text{in}\,\,\langle
e\rangle^{\perp}.\quad$

(Here $D^{2}_{e}$ is given by (29).) This is the penalized cell problem
associated with a uniformly elliptic operator with quasi-periodic
coefficients. Therefore, by [15, Lemma 9.1], there is a constant
$F_{e}^{\perp}(X,y)\in\mathbb{R}$ such that

$\lim_{\delta\to
0^{+}}\sup\left\\{|\tilde{V}^{\delta}_{y}(x^{\prime})+F_{e}^{\perp}(X,y)|\,\mid\,x^{\prime}\in\langle
e\rangle^{\perp}\right\\}=0.$

Notice that $y\mapsto F_{e}^{\perp}(X,y)$ is well-defined and only varies in
the $e$ direction. Indeed, if $k\in\mathbb{Z}^{d}$, then
$\tilde{V}^{\delta}_{y+k}=\tilde{V}^{\delta}_{y}$ so
$F_{e}^{\perp}(X,y+k)=F_{e}^{\perp}(X,y)$. To see that
$F_{e}^{\perp}(X,y)=F_{e}^{\perp}(X,\langle y,e\rangle e)$, observe that

$-F_{e}^{\perp}(X,y)=\lim_{\delta\to
0^{+}}\delta\tilde{V}^{\delta}_{y}(0)=\lim_{\delta\to
0^{+}}\delta\tilde{V}^{\delta}_{\langle y,e\rangle e}(y-\langle y,e\rangle
e)=-F_{e}^{\perp}(X,\langle y,e\rangle e).$

This proves $F_{e}^{\perp}(X,\cdot)$ varies only in the $e$ direction.

It remains to prove the uniform convergence of $(\delta
V^{\delta})_{\delta>0}$. If $e\in\mathbb{R}\mathbb{Z}^{d}$, then (33) is a
periodic homogenization problem in $\langle e\rangle^{\perp}$. (Here the
coefficients are invariant under the group $M_{e}=\mathbb{Z}^{d}\cap\langle
e\rangle^{\perp}$, which is a rank $(d-1)$-subgroup of $\langle
e\rangle^{\perp}$.) Thus, arguing as in [15, Section 8], we see that the rate
of convergence of $(\delta\tilde{V}^{\delta}_{y})_{\delta>0}$ is uniform with
respect to $y$. From this, it is immediate that $(\delta
V^{\delta})_{\delta>0}$ converges uniformly in $\mathbb{T}^{d}$.

On the other hand, if $e\notin\mathbb{R}\mathbb{Z}^{d}$, then the density of
each leaf of the foliation $\\{\mathbb{T}^{d-1}_{e}(r)\\}_{r\in[0,r_{e})}$
implies that, for each $y^{\prime}\in\mathbb{T}^{d}$,

$\sup\left\\{|\delta
V^{\delta}(y)+F_{e}^{\perp}(X,y^{\prime})|\,\mid\,y\in\mathbb{T}^{d}\right\\}=\sup\left\\{|\delta\tilde{V}^{\delta}_{y^{\prime}}(x^{\prime})+F_{e}^{\perp}(X,y^{\prime})|\,\mid\,x^{\prime}\in\langle
e\rangle^{\perp}\right\\}.$

Thus, $\lim_{\delta\to 0^{+}}\delta
V^{\delta}(y)=-F_{e}^{\perp}(X,y^{\prime})$ uniformly in $\mathbb{T}^{d}$, no
matter the choice of $y^{\prime}$. Note that this shows
$F_{e}^{\perp}(X,\cdot)$ is a constant in this case. ∎

Now we prove the properties of $\overline{F}(e,\cdot)$ claimed in Theorem 7:

######  Corollary 4.

$\overline{F}(e,\cdot):\mathcal{S}_{d}\to\mathbb{R}$ is continuous, it
satisfies (7), $|\overline{F}(e,X)|\leq\Lambda\|X\|$, and

$\overline{F}(e,X)=\overline{F}(e,\tilde{X}_{e})\quad\text{if}\,\,X\in\mathcal{S}_{d}.$

###### Proof.

By the proof of Theorem 8, $\tilde{X}_{e}\mapsto\overline{F}(e,\tilde{X}_{e})$
is the homogenized operator associated with the (e.g. Dirichlet)
homogenization problem

$\left\\{\begin{array}[]{r
l}-F(e,D^{2}_{e}U^{\epsilon},\epsilon^{-1}x^{\prime})=0&\text{in}\,\,\langle
e\rangle^{\perp}\\\ U^{\epsilon}=G&\text{on}\,\,\langle
e\rangle^{\perp}\end{array}\right.$

Therefore, the claims follow directly from [16].∎

Finally, we define the homogenized operator
$\overline{F}:(\mathbb{R}^{d}\setminus\mathbb{R}\mathbb{Z}^{d})\times\mathcal{S}_{d}\to\mathbb{R}$
by

(34) $\overline{F}(p,X)=\|p\|\overline{F}(\hat{p},\|p\|^{-1}X).$

###  4.2. Regularity Estimates

We now turn to the regularity of the penalized correctors
$(V^{\delta})_{\delta>0}$ when the operator $F$ is sufficiently regular. To
start with, we consider the two dimensional case since it benefits from
special structure. We then turn to the operators of Theorem 2, which admit
more-or-less smooth penalized correctors due to the strong assumptions imposed
on the matrix field $a$.

Here is the result when $d=2$:

######  Proposition 2.

Let $d=2$ and assume that
$F(e,\cdot):\mathcal{S}_{d}\times\mathbb{T}^{d}\to\mathbb{R}$ is twice
continuously differentiable. Given $\delta>0$, $e\in S^{1}$, and
$X\in\mathcal{S}_{2}$, the penalized corrector $V^{\delta}$ solving (30) is in
$C^{2}(\mathbb{T}^{2})$.

###### Proof.

Let $\\{\tilde{V}^{\delta}_{y}\\}_{y\in\mathbb{T}^{2}}$ be the functions
defined in (32). Since $|F(e,X,y)|\leq\Lambda\|X\|$, it follows that
$\|V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}\leq\Lambda\delta^{-1}\|X\|$. From
this, a straightforward computation involving (7) (and the fact that $\langle
e\rangle^{\perp}$ is one-dimensional) shows that

$-\frac{2\Lambda\|X\|}{\lambda}\leq-D^{2}_{e}\tilde{V}_{y}^{\delta}\leq\frac{2\Lambda\|X\|}{\lambda}\quad\text{in
the viscosity sense in}\,\,\langle e\rangle^{\perp}.$

Hence $\\{\tilde{V}^{\delta}_{y}\\}_{y\in\mathbb{T}^{2}}$ are bounded in
$C^{1,1}$. The equation uniquely and continuously determines the functions
$\\{D^{2}_{e}\tilde{V}^{\delta}_{y}\\}_{y\in\mathbb{T}^{2}}$ by (7) so
actually we are working with $C^{2}$ functions.

Manipulating difference quotients and invoking assumptions (7) and the $C^{2}$
assumption on $F$, we apply the Krylov-Safonov Theorem to find that
$\\{\tilde{V}^{\delta}_{y}\\}_{y\in\mathbb{T}^{2}}\subseteq
C^{2,\alpha}(\langle e\rangle^{\perp})$.

Next, we show that $y\mapsto\tilde{V}^{\delta}_{y}$ is $C^{2,\alpha}$. Fix
$y^{\prime}\in\langle e\rangle^{\perp}$. Using difference quotients again, we
see that the function $\frac{\partial\tilde{V}^{\delta}_{y}}{\partial y_{e}}$
given by

$\frac{\partial\tilde{V}^{\delta}_{y}}{\partial y_{e}}(x^{\prime})=\lim_{h\to
0}\frac{\tilde{V}^{\delta}_{y+he}(x^{\prime})-\tilde{V}^{\delta}_{y}(x^{\prime})}{h}\quad(x^{\prime}\in\langle
e\rangle^{\perp})$

is solution of the uniformly elliptic linear PDE

$\delta\frac{\partial\tilde{V}^{\delta}_{y}}{\partial
y_{e}}-D_{A}F(e,\tilde{X}_{e}+D^{2}_{e}\tilde{V}^{\delta}_{y},y+x^{\prime}):D^{2}_{e}\left(\frac{\partial\tilde{V}^{\delta}_{y}}{\partial
y_{e}}\right)-\langle
D_{y}F(e,\tilde{X}_{e}+D^{2}_{e}\tilde{V}^{\delta}_{y},y+\tilde{x}),e\rangle=0\quad\text{in}\,\,\langle
e\rangle^{\perp}.$

Thus, $\frac{\partial\tilde{V}^{\delta}_{y}}{\partial y_{e}}$ is $C^{2}$ in
$\langle e\rangle^{\perp}$ and the uniform ellipticity gives a constant $C>0$
such that

$\displaystyle\sup\left\\{\delta\left\|\frac{\partial\tilde{V}^{\delta}_{y}}{\partial
y_{e}}\right\|_{L^{\infty}(\langle
e\rangle^{\perp})}+\left\|D^{2}\left(\frac{\partial\tilde{V}^{\delta}_{y}}{\partial
y_{e}}\right)\right\|_{L^{\infty}(\langle
e\rangle^{\perp})}\,\mid\,y\in\mathbb{T}^{2}\right\\}\leq C.$

Differentiating again, we find that the second derivative
$\frac{\partial^{2}\tilde{V}^{\delta}_{y}}{\partial y_{e}^{2}}$ (defined
anlogously) is $C^{2}$ and satisfies

$\sup\left\\{\delta\left\|\frac{\partial^{2}\tilde{V}^{\delta}_{y}}{\partial
y_{e}^{2}}\right\|_{C(\langle
e\rangle^{\perp})}+\left\|D^{2}\left(\frac{\partial^{2}\tilde{V}^{\delta}_{y}}{\partial
y_{e}^{2}}\right)\right\|_{L^{\infty}(\langle
e\rangle^{\perp})}\,\mid\,y\in\mathbb{T}^{2}\right\\}\leq C\delta^{-1}.$

Finally, observe that if $y\in\mathbb{T}^{d}$ and $e^{\prime}\in
S^{d-1}\cap\langle e\rangle^{\perp}$, then

$\displaystyle D^{2}_{e}V^{\delta}(y)=D^{2}_{e}\tilde{V}^{\delta}_{y}(0),$
$\displaystyle\quad\langle
D^{2}V^{\delta}(y)e,e\rangle=\frac{\partial^{2}\tilde{V}^{\delta}_{y}}{\partial
y_{e}^{2}}(0),$ $\displaystyle\langle D^{2}V^{\delta}(y)e,e^{\prime}\rangle$
$\displaystyle=\left\langle
D\left(\frac{\partial\tilde{V}^{\delta}_{y}}{\partial
y_{e}}\right)(0),e^{\prime}\right\rangle.$

Therefore, the previous considerations show that $V^{\delta}\in
C^{2}(\mathbb{T}^{d})$. ∎

Finally, in the quasi-linear set-up, we have

######  Proposition 3.

If $a$ and $m$ satisfy the assumptions of Theorem 2 and
$F(p,X,y)=m(y,\hat{p})^{-1}\text{tr}(a(y,\hat{p})\tilde{X}_{\hat{p}})$, then,
for each $\delta>0$, the penalized corrector $V^{\delta}$ of (30) satisfies
$V^{\delta}\in C^{2,\alpha}(\mathbb{T}^{d})$.

###### Proof.

The proof proceeds as in the last proposition. To obtain $C^{2,\alpha}$
estimates, we use Schauder estimates for linear elliptic equations (cf. [31,
Chapter 6] or [13, Chapter 8]). ∎

###  4.3. Approximate Correctors

This section is devoted to the proof of Theorem 7. In general, in the setting
of Theorem 1, if $F$ is not twice differentiable, we approximate it by a
family of regularized operators $(F^{\mu})_{\mu>0}$ that are. This is made
precise in the next result:

######  Proposition 4.

Given $e\in S^{1}$ and $F$ satisfying the assumptions of Theorem 1, there is a
family $(F_{e}^{\mu})_{\mu>0}$ of operators, twice continuously differentiable
in $\mathcal{S}_{2}\times\mathbb{T}^{2}$, such that, for each $\mu>0$,

(35)
$\displaystyle\lambda\|\tilde{N}^{+}_{e}\|-\Lambda\|\tilde{N}_{e}^{-}\|\leq
F^{\mu}_{e}(M+N,x)-F^{\mu}_{e}(M,x)\leq\Lambda\|\tilde{N}_{e}^{+}\|-\lambda\|\tilde{N}_{e}^{-}\|\quad\text{if}\,\,M,N\in\mathcal{S}_{2},$
(36) $\displaystyle|F_{e}^{\mu}(M,y^{\prime})-F_{e}^{\mu}(M,y)|\leq
C(1+\mu+\|M\|)\|x^{\prime}-x\|\quad\text{if}\,\,M\in\mathcal{S}_{2},\,\,y,y^{\prime}\in\mathbb{T}^{2},$
(37)
$\displaystyle|F_{e}^{\mu}(M,y)-F(e,M,y)|\leq\Lambda\mu+\omega^{e}_{1+\|M\|}(\mu)\quad\text{if}\,\,(M,y)\in\mathcal{S}_{2}\times\mathbb{T}^{2}.$

Here, for each $R>0$, $\omega_{R}^{e}$ is the modulus of $F$ in $\\{e\\}\times
B(0,R)\times\mathbb{T}^{d}$, that is,

$\omega_{R}^{e}(\delta)=\sup\left\\{|F(e,X,y)-F(e,X^{\prime},y^{\prime})|\,\mid\,(X,y),(X^{\prime},y^{\prime})\in
B(0,R)\times\mathbb{T}^{d},\,\,\|y-y^{\prime}\|\leq\delta\right\\}.$

Proposition 4 can be proved by mollifying the operator $(X,y)\mapsto
F(e,X,y)$. The details are left to the interested reader.

###### Proof of Theorem 7.

We will assume that $F$ satisfies the assumptions of Theorem 1. If instead $F$
satisfies the assumptions of Theorem 2, then Proposition 3 shows that the
penalized correctors of Theorem 8 are $C^{2}$ so there is no need to
regularize.

Fix $\nu>0$. Let $\delta,\mu>0$ be free variables and let $V^{\delta,\mu}\in
C^{2}(\mathbb{T}^{2})$ be the solution of (30) with $F$ replaced by $F^{\mu}$.
Recall that

(38)
$\|D^{2}_{e}V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}\leq\frac{2\Lambda\|X\|}{\lambda}.$

Thus, if $y\in\mathbb{T}^{2}$, then

$\displaystyle|\overline{F}(e,X)-F(e,\tilde{X}_{e}+D^{2}_{e}V^{\delta},y)|$
$\displaystyle\leq(I)+(II)+(III),$

where

$\displaystyle(I)$ $\displaystyle=\|\overline{F}^{\mu}(e,X)+\delta
V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})},$ $\displaystyle(II)$
$\displaystyle=|F(e,\tilde{X}_{e}+D^{2}_{e}V^{\delta},y)-F^{\mu}_{e}(\tilde{X}_{e}+D^{2}_{e}V^{\delta},y)|$
$\displaystyle\leq\Lambda\mu+\omega^{e}_{1+2\lambda^{-1}\Lambda\|X\|}(\mu),$
$\displaystyle(III)$
$\displaystyle=|\overline{F}(e,X)-\overline{F}^{\mu}(e,X)|.$

Using Lemma 2 below and (38), it is not hard to show that

$|\overline{F}(e,X)-\overline{F}^{\mu}(e,X)|\leq\|\overline{F}^{\mu}(e,X)+\delta
V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}+\Lambda\mu+\omega^{e}_{1+(1+2\lambda^{-1}\Lambda)\|X\|)}(\mu).$

Indeed, the approximate corrector $V^{\delta,\mu}$ satisfies

$\delta
V^{\delta,\mu}-F^{\mu}(e,\tilde{X}_{e}+D^{2}_{e}V^{\delta,\mu})=0\quad\text{in}\,\,\mathbb{T}^{2}.$

Thus, (38) and the choice of $F^{\mu}$ yield

$|-F(e,\tilde{X}_{e}+D^{2}_{e}V^{\delta,\mu})+\overline{F}^{\mu}(e,X)|\leq\|\overline{F}^{\mu}(e,X)+\delta
V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}+\Lambda\mu+\omega^{e}_{1+(1+2\lambda^{-1}\Lambda)\|X\|}(\mu)\quad\text{in}\,\,\mathbb{T}^{2}.$

Hence, by Lemma 2, we have

$(III)=|\overline{F}(e,X)-\overline{F}^{\mu}(e,X)|\leq\|\overline{F}^{\mu}(e,X)+\delta
V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}+\Lambda\mu+\omega^{e}_{1+(1+2\Lambda)\|X\|}(\mu).$

We conclude that if we first choose $\mu>0$ small enough that

$\Lambda\mu+\omega^{e}_{1+(1+2\lambda^{-1}\Lambda)\|X\|)}(\mu)\leq\frac{\nu}{3}$

and then choose $\delta>0$ so small that $\|\overline{F}^{\mu}(e,X)+\delta
V^{\delta}\|_{L^{\infty}(\mathbb{T}^{2})}\leq\nu/3$, then we obtain

$-\nu\leq\overline{F}(e,X)-F(e,\tilde{X}_{e}+D^{2}V^{\delta}_{e},y)\leq\nu\quad\text{in}\,\,\mathbb{T}^{2}.$

∎

###  4.4. Invariant Measures

This section makes the connection between the homogenized operators
$\overline{F}$ and oscillating functions $F^{\perp}_{e}$ obtained in Theorem 7
and the invariant measures of Theorem 3.

To start with, we give the basic existence result for invariant measures:

######  Proposition 5.

Assume that $a$ satisfies the assumptions of Theorem 2. For each $e\in
S^{d-1}$, $\mathscr{I}^{a}_{e}$ is non-empty. Furthermore, if
$e\in\mathbb{R}\mathbb{Z}^{d}$, then there is an $r_{e}$-periodic function
$\mu_{e}:\mathbb{R}\to\mathscr{I}^{a}_{e}$, $\mu_{e}:s\mapsto\mu_{e}^{s}$,
such that, for each $r\in[0,r_{e})$, the support of $\mu_{e}$ equals
$\mathbb{T}^{d-1}_{e}(r)$ and
$\mu_{e}^{s}\ll\mathcal{H}^{d-1}\restriction_{\mathbb{T}^{d-1}_{e}(r)}$.

###### Proof.

First, assume that $e\in\mathbb{R}\mathbb{Z}^{d}$. Given $r\in[0,r_{e})$, the
trajectories of the SDE (15) with $\langle X^{e}_{0},e\rangle=r$ satisfy
$\langle X^{e}_{t},e\rangle=r$ for all $t>0$. Hence, as long as we restrict to
initial distributions concentrated in $\mathbb{T}^{d-1}_{e}(r)$, we can
consider (15) as a process in $\mathbb{T}^{d-1}_{e}(r)$. Restricted to
$\mathbb{T}^{d-1}_{e}(r)$, (15) is uniformly non-degenerate. Therefore, it has
a unique invariant measure $\mu_{e}^{r}$ that has full support in
$\mathbb{T}^{d-1}_{e}(r)$ and
$\mu_{e}^{r}\ll\mathcal{H}^{d-1}\restriction_{\mathbb{T}^{d-1}_{e}(r)}$ (cf.
[11] or [42]).

If $e\notin\mathbb{R}\mathbb{Z}^{d}$, then there is a sequence
$(e_{n})_{n\in\mathbb{N}}\subseteq S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$ such
that $e=\lim_{n\to\infty}e_{n}$. By the Banach-Alaoglu Theorem,
$(\mu_{e_{n}}^{0})_{n\in\mathbb{N}}$ has a weak-$*$ limit point $\mu$. A quick
argument shows that $\mu\in\mathscr{I}^{a}_{e}$ necessarily holds. Further,
Lemma 3 in the appendix shows that $\mu\ll\mathcal{L}^{d}$. ∎

We still need to understand the structure of the sets
$\\{\mathscr{I}^{a}_{e}\\}_{e\in S^{d-1}}$. Here and in Section 7 it will be
convenient to use the following variant of Theorem 7:

######  Proposition 6.

If $a$ satisfies the assumptions of Theorem 2 and $f\in C(\mathbb{T}^{d})$,
then, for each $e\in S^{d-1}$ and $\delta>0$, there is a unique $V^{\delta}\in
C(\mathbb{T}^{d})$ solving the degenerate elliptic PDE:

(39) $\delta
V^{\delta}-\text{tr}(a(y,e)D^{2}_{e}V^{\delta})=f(y)\quad\text{in}\,\,\mathbb{T}^{d}.$

Furthermore, there is an $f^{\perp}_{e}\in C(\mathbb{T}^{d})$ varying only in
the $e$ direction such that

$\lim_{\delta\to 0^{+}}\|\delta
V^{\delta}-f^{\perp}_{e}\|_{L^{\infty}(\mathbb{T}^{d})}=0.$

If $e\notin\mathbb{R}\mathbb{Z}^{d}$, then there is a constant
$\bar{f}(e)\in\mathbb{R}$ such that $f^{\perp}_{e}\equiv\bar{f}(e)$.

Finally, if $f\in C^{2,\alpha}(\mathbb{T}^{d})$, then $V^{\delta}\in
C^{2,\alpha}(\mathbb{T}^{d})$.

This readily leads to the proof of Theorem 3. In addition, we will show that
the operator $F^{\perp}_{e}$ of Theorem 7 is determined in this case by the
formula

(40)
$F^{\perp}_{e}(X,y)=m^{\perp}_{e}(y)^{-1}\text{tr}\left(a^{\perp}_{e}(y)\tilde{X}_{e}\right),$

where $a^{\perp}_{e}$ is given by (19) and $m^{\perp}_{e}$ is defined
analogously.

###### Proof of Theorem 3.

We start with (i). Assume that $e\notin\mathbb{R}\mathbb{Z}^{d}$. By the Riesz
Representation Theorem, we can define a Borel probability measure
$\bar{\mu}_{e}$ in $\mathbb{T}^{d}$ by

$\int_{\mathbb{T}^{d}}f(y)\bar{\mu}_{e}(dy)=\bar{f}(e),$

where $\bar{f}(e)$ is the constant in Theorem 6. We claim that
$\mathscr{I}^{a}_{e}=\\{\bar{\mu}_{e}\\}$.

Indeed, if $\mu\in\mathscr{I}^{a}_{e}$, $f\in C^{2,\alpha}(\mathbb{T}^{d})$,
and $V^{\delta}$ is the associated solution of (39), then

$\int_{\mathbb{T}^{d}}\delta
V^{\delta}(y)\,\mu(dy)=\int_{\mathbb{T}^{d}}f(y)\,\mu(dy).$

Sending $\delta\to 0^{+}$, we find
$\int_{\mathbb{T}^{d}}f(y)\,\mu(dy)=\bar{f}(e)$. This proves
$\mu=\bar{\mu}_{e}$ by definition. Since $\mathscr{I}^{a}_{e}$ is non-empty by
Proposition 5, it also shows that $\bar{\mu}_{e}\in\mathscr{I}^{a}_{e}$.

Next, we turn to (ii). It remains to show that $\mathscr{I}^{a}_{e}$ is the
closed convex hull of $\\{\mu_{e}^{s}\,\mid\,s\in\mathbb{R}\\}$. By the Krein-
Milman Theorem, we only have to prove that the latter equals the set of
extreme points.

First, notice that if $f\in C(\mathbb{T}^{d})$, $(V^{\delta})_{\delta>0}$ are
the associated solutions of (39), and $s\in[0,r_{e})$, then Proposition 6 and
the definition of $\mathscr{I}^{a}_{e}$ give

$\displaystyle\int_{\mathbb{T}^{d}}f(y)\,\mu_{e}^{s}(dy)$
$\displaystyle=\lim_{\delta\to 0^{+}}\int_{\mathbb{T}^{d}}\left(\delta
V^{\delta}(y)-\text{tr}\left(A(y,e)D^{2}V^{\delta}(y)\right)\right)\,\mu_{e}^{s}(dy)$
$\displaystyle=\lim_{\delta\to 0^{+}}\int_{\mathbb{T}^{d}}\delta
V^{\delta}(y)\,\mu_{e}^{s}(dy)=f_{e}^{\perp}(se).$

Now assume $\mu$ is an extreme point of $\mathcal{I}_{e}$. Repeating the
previous computation with $\mu^{s}_{e}$ replaced by $\mu$, we find, for each
$f\in C(\mathbb{T}^{d})$,

$\int_{\mathbb{T}^{d}}f(y)\,\mu(dy)=\int_{\mathbb{T}^{d}}f_{e}^{\perp}(\langle
y,e\rangle
e)\,\mu(dy)=\int_{\mathbb{T}^{d}}\left(\int_{\mathbb{T}^{d}}f(y^{\prime})\,\mu_{e}^{\langle
y,e\rangle}(dy^{\prime})\right)\,\mu(dy).$

In particular, letting $\nu\in\mathscr{P}(r_{e}\mathbb{T})$ be the push-
forward of $\mu$ onto $r_{e}\mathbb{T}$ given by

$\nu(A)=\mu(\\{y\in\mathbb{T}^{d}\,\mid\,\langle y,e\rangle\in A\\}),$

we obtain $\mu=\int_{r_{e}\mathbb{T}}\mu_{e}^{s}\,\nu(ds)$. Now notice that if
$A\subseteq r_{e}\mathbb{T}$ and $\nu(A)>0$, then

$\mu=\nu(A)\fint_{A}\mu_{e}^{s}\,\nu(ds)+(1-\nu(A))\fint_{A^{c}}\mu_{e}^{s}\,\nu(ds).$

Hence, since $\mu$ is an extreme point of $\mathcal{I}_{e}$, either $\nu(A)=1$
or $\nu(A)=0$. From this, an elementary argument shows that
$\nu=\delta_{s^{\prime}}$ for some $s^{\prime}\in[0,r_{e})$. In particular,
$\mu=\mu_{e}^{s^{\prime}}$. This proves
$\\{\mu_{e}^{s}\,\mid\,s\in\mathbb{R}\\}$ equals the set of extreme points of
$\mathscr{I}^{a}_{e}$.

Finally, we prove (iii). If $m\equiv 1$ in Theorem 7, $\delta>0$, and
$X\in\mathcal{S}_{d}$ are arbitrary, then the solution $V^{\delta}$ of (30)
solves (39) with $f(y)=\text{tr}(a(y,e)X)$. Hence

$\displaystyle\overline{F}(e,X)$
$\displaystyle=\int_{\mathbb{T}^{d}}\text{tr}(a(y,e)X)\,\bar{\mu}_{e}(dy)\quad\text{if}\,\,e\notin\mathbb{R}\mathbb{Z}^{d},$
$\displaystyle F^{\perp}_{e}(X,y)$
$\displaystyle=\int_{\mathbb{T}^{d}}\text{tr}(a(y^{\prime},e)X)\,\mu_{e}^{\langle
y,e\rangle}(dy^{\prime})\quad\text{if}\,\,e\in\mathbb{R}\mathbb{Z}^{d}.$

This shows that the operator $\overline{F}$ is given by (14) with $\bar{a}$
and $a^{\perp}_{e}$ as in (17) and (19).

When $m\not\equiv 1$, the solution $V^{\delta}$ of (30) is also solution of

$\delta
m(y,e)V^{\delta}-\text{tr}\left(a(y,e)D^{2}_{e}V^{\delta}\right)=\text{tr}\left(a(y,e)X\right)\quad\text{in}\,\,\mathbb{T}^{d}.$

Given $\epsilon>0$, for small enough $\delta$, this yields

$|F^{\perp}_{e}(X,y)m(y,e)-\text{tr}\left(a(y,e)D^{2}_{e}V^{\delta}\right)-\text{tr}\left(a(y,e)X\right)|\leq\epsilon\quad\text{in}\,\,\mathbb{T}^{d}.$

When $e\in\mathbb{R}\mathbb{Z}^{d}$, we integrate with respect to
$\mu_{e}^{s}$ for a given $s\in[0,r_{e})$ to find

$\left|F^{\perp}_{e}(X,se)\int_{\mathbb{T}^{d}}m(y^{\prime},e)\,\mu_{e}^{s}(dy^{\prime})-\text{tr}\left(a^{\perp}_{e}(se)X\right)\right|\leq\epsilon.$

The arbitrariness of $\epsilon$ implies (40). A similar computation applies in
the case that $e\notin\mathbb{R}\mathbb{Z}^{d}$, giving (14). ∎

###  4.5. Correctors in Rational Directions

Finally, we build correctors of the cell problem when
$e\in\mathbb{R}\mathbb{Z}^{d}$ and study the regularity of the oscillating
function $f_{e}^{\perp}$. These will be convenient in the analysis of the
forced problem (20) as well as the proof of Corollary 1 concerning generic
discontinuities.

This section is devoted to the study of the following cell problem:

(41)
$-\text{tr}\left(a(y,e)D^{2}_{e}\tilde{V}_{e}\right)=f(y)-f_{e}^{\perp}(\langle
y,e\rangle e)\quad\text{in}\,\,\mathbb{T}^{d}.$

Notice that if a solution exists, then $f_{e}^{\perp}$ is necessarily as in
Proposition 6.

######  Proposition 7.

If $a$ satisfies the assumptions of Theorem 2 and $f\in
C^{2,\alpha}(\mathbb{T}^{d})$, then there is a solution $V_{e}\in
C^{2,\alpha}(\mathbb{T}^{d})$ of (41) and $f_{e}^{\perp}\in
C^{2,\alpha}(\mathbb{T}^{d})$. Further, if $\overline{V}_{e}\in
C^{2,\alpha}(\mathbb{T}^{d})$ is any other solution of (41), then
$V_{e}-\overline{V}_{e}$ varies only in the $e$ direction.

Note that this implies $a^{\perp}_{e}$ and $m^{\perp}_{e}$ of (40) are both
$C^{2,\alpha}$.

###### Proof of Proposition 7.

To start with, let $\varphi\in C^{2,\alpha}([0,r_{e}])$ be a function with
$\varphi(0)=0$. For each $s\in[0,r_{e})$, let
$\tilde{V}_{se}:\mathbb{T}^{d-1}_{e}(0)\to\mathbb{R}$ be the solution of the
cell problem

$\left\\{\begin{array}[]{r
l}-\text{tr}\left(a(se+x^{\prime})D^{2}_{e}\tilde{V}_{se}\right)=f(se+x^{\prime})-f_{e}^{\perp}(se)&\text{in}\,\,\mathbb{T}^{d-1}_{e}(0)\\\
\tilde{V}_{se}(0)=\varphi(s)\end{array}\right.$

Notice that $\tilde{V}_{se}$ is unique.

Step 1: Preliminary regularity of $f_{e}^{\perp}$

We claim that there is a constant $C>0$ such that

$|f_{e}^{\perp}((s+h)e)-f_{e}^{\perp}(se)|\leq
C|h|\quad\text{if}\,\,s,h\in\mathbb{R}.$

Indeed, first, recall (cf. [23, Lemma 3.1] or [17, Lemma 2.1]) that there is a
$C_{*}>0$ depending only on $\lambda,\Lambda$, and
$\|f\|_{L^{\infty}(\mathbb{T}^{d})}$ such that

$\sup\left\\{\|\tilde{V}_{se}-\tilde{V}_{se}(0)\|_{L^{\infty}(\mathbb{T}^{d-1}_{e}(0))}\,\mid\,s\in[0,r_{e})\right\\}\leq
C_{*}.$

Therefore, by Schauder estimates,

$C_{0}:=\sup\left\\{\|D^{2}_{e}\tilde{V}_{se}\|_{L^{\infty}(\mathbb{T}^{d-1}_{e}(0))}\,\mid\,s\in[0,r_{e})\right\\}<\infty.$

Therefore, by (9), given $s\in[0,r_{e})$, the function $\tilde{V}_{(s+h)e}$
satisfies

$\displaystyle-f(se+x^{\prime})-\text{tr}\left(a(se+x^{\prime})D^{2}_{e}\tilde{V}_{(s+h)e}\right)$
$\displaystyle\leq-
f_{e}^{\perp}((s+h)e)+(C_{0}\|a(\cdot,e)\|_{C^{1,\alpha}(\mathbb{T}^{d})}+\|f\|_{C^{1,\alpha}(\mathbb{T}^{d})})|h|$
$\displaystyle-f(se+x^{\prime})-\text{tr}\left(a(se+x^{\prime})D^{2}_{e}\tilde{V}_{(s+h)e}\right)$
$\displaystyle\geq-
f_{e}^{\perp}((s+h)e)-\left(C_{0}\|a(\cdot,e)\|_{C^{1,\alpha}(\mathbb{T}^{d})}+\|f\|_{C^{1,\alpha}(\mathbb{T}^{d})}\right)|h|$

From this and Lemma 2, we find

$|f_{e}^{\perp}((s+h)e)-f_{e}^{\perp}(se)|\leq\left(C_{0}\|a(\cdot,e)\|_{C^{1,\alpha}(\mathbb{T}^{d})}+\|f\|_{C^{1,\alpha}(\mathbb{T}^{d})}\right)|h|$

Step 2: Convenient extension of $\\{\tilde{V}_{se}\\}_{s\in[0,r_{e})}$

At this stage, it is more-or-less inevitable to extend the function
$s\mapsto\tilde{V}_{se}$ from $[0,r_{e})$ to $\mathbb{R}$. Given
$r\in\mathbb{R}$, define $\tilde{V}_{re}\in C(\mathbb{T}^{d-1}_{e}(0))$ by

$\tilde{V}_{re}(x^{\prime})=\tilde{V}_{(r-Nr_{e})e}(x^{\prime}+Nr_{e}e)\quad\text{if}\,\,0\leq
r-Nr_{e}<r_{e}\,\,\text{and}\,\,N\in\mathbb{Z}.$

We refine the choice of $\varphi$ to make $r\mapsto\tilde{V}_{re}$ continuous.
Since it coincides with $\varphi$, $r\mapsto\tilde{V}_{re}(0)$ is
differentiable at each $r\in(0,r_{e})$. To get a derivative at $0$ or $r_{e}$,
we constrain $\varphi$ so that the following identity holds:

$\varphi(r_{e})=\tilde{V}_{0}(r_{e}e).$

Note this isn’t circular since $\tilde{V}_{0}$ is determined by $\varphi(0)$
alone. We leave it to the reader to check that now $r\mapsto\tilde{V}_{re}$
takes $\mathbb{R}$ continuously into $C(\mathbb{T}^{d-1}_{e}(0))$.

Step 3: Differentiability in $e$ direction

To get a derivative of $r\mapsto\tilde{V}_{re}$, we further refine the choice
of $\varphi$. First of all, notice that, with the Lipschitz estimate on
$f_{e}^{\perp}$ in hand, we deduce that there is a $C_{1}>0$ such that, for
each $s\in[0,r_{e})$ and $h>0$ small enough, the function
$\tilde{W}^{h}_{se}=\tilde{V}_{(s+h)e}-\tilde{V}_{se}$ satisfies

$\displaystyle\left\\{\begin{array}[]{r
l}-C_{1}|h|\leq-\text{tr}\left(a(se+x^{\prime})D^{2}_{e}\tilde{W}^{h}_{se}\right)\leq
C_{1}|h|&\text{in}\,\,\mathbb{T}^{d-1}_{e}(0)\\\
\tilde{W}^{h}_{se}(0)=\varphi(s+h)-\varphi(s)\end{array}\right.$

From this, the compactness of $\mathbb{T}^{d-1}_{e}(0)$, and the Krylov-
Safonov Theorem, it follows that there is a $B>0$ depending only on $C_{1}$
such that

$\|\tilde{W}^{h}_{se}\|_{L^{\infty}(\mathbb{T}^{d-1}_{e}(0))}\leq B|h|.$

We caim that $\tilde{W}_{se}=\lim_{h\to 0^{+}}h^{-1}\tilde{W}^{h}_{se}$ exists
in $C(\mathbb{T}^{d})$ for each $s\in[0,r_{e})$. Indeed, suppose
$(h_{n})_{n\in\mathbb{N}}\subseteq(0,\infty)$ is a sequence such that
$\lim_{n\to\infty}h_{n}=0$ and $h_{n}^{-1}\tilde{W}^{h_{n}}_{se}\to\tilde{W}$
uniformly in $\mathbb{T}^{d-1}_{e}(0)$ for some $\tilde{W}\in
C(\mathbb{T}^{d-1}_{e}(0))$. Passing to a subsequence, if necessary, we can
assume that there is a $\tilde{c}\in\mathbb{R}$ such that

$\tilde{c}=\lim_{n\to\infty}\frac{f_{e}^{\perp}((s+h_{n})e)-f_{e}^{\perp}(se)}{h_{n}}.$

It follows that $\tilde{W}$ is a viscosity solution of

$-\langle
Df(se+x^{\prime}),e\rangle-\text{tr}\left(a(se+x^{\prime},e)D^{2}_{e}\tilde{W}\right)=-\tilde{c}+\text{tr}\left(\langle
D_{y}a(se+x^{\prime},e),e\rangle
D^{2}_{e}\tilde{V}_{se}\right)\quad\text{in}\,\,\mathbb{T}^{d-1}_{e}(0).$

By Lemma 2, $\tilde{c}$ does not depend on the sequence
$(h_{n})_{n\in\mathbb{N}}$, and it follows from the normalization
$\tilde{W}(0)=\varphi^{\prime}(s)$ that $\tilde{W}$ is also unique. In
particular, there is a function $\tilde{W}_{se}$ such that, for each
$s\in[0,r_{e})$,

$h^{-1}(\tilde{V}_{(s+h)e}-\tilde{V}_{se})\to\tilde{W}_{se}\quad\text{uniformly
in}\,\,\mathbb{T}^{d-1}_{e}(0)\,\,\text{as}\,\,h\to 0^{+}.$

Note that $\tilde{W}_{se}$ is the unique viscosity solution of

$-\langle
Df(se+x^{\prime}),e\rangle-\text{tr}\left(a(se+x^{\prime})D^{2}_{e}\tilde{W}_{se}\right)=-\langle
Df_{e}^{\perp}(se),e\rangle+\text{tr}\left(\langle
D_{y}a(se+x^{\prime}),e\rangle
D^{2}_{e}\tilde{V}_{se}\right)\quad\text{in}\,\,\mathbb{T}^{d-1}_{e}(0).$

Note that above we considered one-sided derivatives. It is straightforward to
see that, for $s\in(0,r_{e})$, the restriction $h>0$ was unnecessary. To get a
geniune (two-sided) derivative at $0$ or $r_{e}$, however, we need to add a
compatibility condition to $\varphi$. We require the following one:

(42) $\varphi^{\prime}(r_{e})=\tilde{W}_{0}(r_{e}e).$

We leave it to the reader to verify that (42) implies that
$s\mapsto\tilde{V}_{se}$ is differentiable at each $s\in[0,r_{e}]$. In fact,
it implies that $s\mapsto\tilde{V}_{se}$ is differentiable in $\mathbb{R}$
with

$\lim_{h\to
0}\frac{\tilde{V}_{(s+h)e}-\tilde{V}_{se}}{h}=\tilde{W}_{(s-Nr_{e})e}(\cdot+Nr_{e}e)\quad\text{uniformly
in}\,\,\mathbb{T}^{d}\quad\text{if}\,\,0\leq s-Nr_{e}<r_{e}.$

Furthermore, $\|\tilde{W}_{se}\|_{L^{\infty}(\mathbb{T}^{d-1}_{e}(0))}\leq B$
independent of $s\in\mathbb{R}$.

Finally, notice that if we define $V_{e}:\mathbb{T}^{d}\to\mathbb{R}$ by

$V_{e}(x)=\tilde{V}_{\langle x,e\rangle e}(x-\langle x,e\rangle e)$

then $V_{e}$ solves (41) and

$\langle DV_{e}(x),e\rangle=\tilde{W}_{\langle x,e\rangle e}(x-\langle
x,e\rangle e)\quad\text{if}\,\,x\in\mathbb{T}^{d}.$

In particular, $V_{e}\in C^{1}(\mathbb{T}^{d})$.

Step 4: Second derivative in $e$ direction Since
$\tilde{W}_{se}(0)=\varphi^{\prime}(s)$ for all $s\in[0,r_{e})$ by
construction, by imposing compatibility conditions on
$\varphi^{\prime\prime}$, we can repeat the previous arguments to show that
$V_{e}\in C^{2,\alpha}(\mathbb{T}^{d})$. (The Hölder continuity of
$D^{2}f_{e}^{\perp}$ follows by arguing as in the proof that it is Lipschitz;
from this, the linearity of the equation can be used to show $\langle
D^{2}V_{e}e,e\rangle$ is Hölder continuous with respect to the $e$ variable.)
∎

##  5\. Homogenization in Irrational Directions

In this section, we undertake the main step in the proofs of Theorems 1 and 2.
We prove that the solutions $(u^{\epsilon})_{\epsilon>0}$ are described in the
limit $\epsilon\to 0^{+}$ by the homogenized equation at any contact point
where the level set of the normal vector is irrational. It turns out that once
this is proved, the remaining viscosity inequalities follow directly.

###  5.1. Solutions in Irrational Directions

Due to the difficulty analyzing (1) at contact points with rational normals,
we are led to consider an a priori weaker notion of viscosity solution. In
what follows, we are interested in functions $\tilde{u}$ that satisfy one or
more of the viscosity inequalities

(43) $\tilde{u}_{t}-\overline{F}^{*}(D\tilde{u},D^{2}\tilde{u})\leq
0,\quad\tilde{u}_{t}-\overline{F}_{*}(D\tilde{u},D^{2}\tilde{u})\geq 0.$

The difficulties encountered in rational directions motivate the following
definition:

######  Definition 1.

Given an open set $U\subseteq\mathbb{R}^{d}\times(0,\infty)$, we say that a
locally bounded, upper semi-continuous function
$\tilde{u}:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a _subsolution of (
43) in irrational directions in $U$_ if there is a $K>0$ such that, given a
smooth function $\varphi:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ such
that $\tilde{u}-\varphi$ has a strict local maximum at $(x_{0},t_{0})\in U$,
the following conditions are met:

  * (a)

If $D\varphi(x_{0},t_{0})\in\mathbb{R}^{d}\setminus\mathbb{R}\mathbb{Z}^{d}$,
then

$\varphi_{t}(x_{0},t_{0})-\overline{F}^{*}(D\varphi(x_{0},t_{0}),D^{2}\varphi(x_{0},t_{0}))\leq
0.$

  * (b)

If $D\varphi(x_{0},t_{0})\in\mathbb{R}\mathbb{Z}^{d}\setminus\\{0\\}$ and

$\left\|\left(\text{Id}-\widehat{D\varphi}(x_{0},t_{0})\otimes\widehat{D\varphi}(x_{0},t_{0})\right)D^{2}\varphi(x_{0},t_{0})\right\|\leq\delta\|D\varphi(x_{0},t_{0})\|,$

then

$\varphi_{t}(x_{0},t_{0})\leq K\delta\|D\varphi(x_{0},t_{0})\|.$

  * (c)

If $\|D\varphi(x_{0},t_{0})\|=\|D^{2}\varphi(x_{0},t_{0})\|=0$, then

$\varphi_{t}(x_{0},t_{0})\leq 0.$

Similarly, a locally bounded, lower semi-continuous function
$\tilde{v}:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a _supersolution of
( 43) in irrational directions in $U$_ if $-\tilde{v}$ satisfies the
definition of subsolution in irrational directions with $\overline{F}_{*}$
replacing $\overline{F}^{*}$ in (a). A locally bounded, continuous function in
$\mathbb{R}^{d}\times(0,\infty)$ is a _solution of ( 43) in irrational
directions in $U$_ if it is both a subsolution and a supersolution.

Combining this definition with the definition of viscosity solution of Barles
and Georgelin [6], we can prove that a sub- or supersolution of (43) in
irrational directions is actually a sub- or supersolution in the usual sense.
Let us first recall the definition of [6]:

######  Definition 2.

Given an open set $U\subseteq\mathbb{R}^{d}\times(0,\infty)$, we say that a
locally bounded, upper semi-continuous function
$\tilde{u}:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a viscosity
subsolution of (43) in $U$ if, for each smooth function
$\varphi:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ and each point
$(x_{0},t_{0})\in U$ such that $\tilde{u}-\varphi$ has a strict local maximum
at $(x_{0},t_{0})$, the following inequalities hold:

  * (i)

If $D\varphi(x_{0},t_{0})\neq 0$, then

$\varphi_{t}(x_{0},t_{0})-\overline{F}^{*}(D\varphi(x_{0},t_{0}),D^{2}\varphi(x_{0},t_{0}))\leq
0.$

  * (ii)

If $\|D\varphi(x_{0},t_{0})\|=\|D^{2}\varphi(x_{0},t_{0})\|=0$, then

$\varphi_{t}(x_{0},t_{0})\leq 0.$

Analogously, we say that a locally bounded, lower semi-continuous function
$\tilde{v}:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a viscosity
supersolution of (43) in $U$ if $-\tilde{v}$ is a viscosity subsolution in $U$
with $\overline{F}_{*}$ replacing $\overline{F}^{*}$. A locally oubnded,
continuous function in $\mathbb{R}^{d}\times(0,\infty)$ is a viscosity
solution of (43) if it is both a sub- and supersolution.

The fact that this definition is equivalent to the usual one is proved arguing
exactly as in [6, Proposition 2.2].

Using Definition 2 and the comparison machinery for viscosity solutions of
second order parabolic equations, we can prove the following:

######  Theorem 9.

Given $U\subseteq\mathbb{R}^{d}\times(0,\infty)$, if
$\tilde{u}:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a subsolution
(resp. supersolution) of (43) in irrational directions in $U$, then it is a
viscosity subsolution (resp. supersolution) in $U$.

###### Proof.

Modulo cosmetic changes, this is precisely [39, Theorem 2]. ∎

###  5.2. Homogenization

In view of the previous theorem, to prove Theorem 2, we only need to show that
the half-relaxed limits $\bar{u}^{*}$ and $\bar{u}_{*}$ are sub- and
supersolution, respectively, of (43) in irrational directions. In particular,
this eliminates any non-trivial analysis of the behavior of those functions at
contact points where the normal is rational.

At contact points where the normal vector is irrational, we will use the
approximate correctors of the previous section and Evans’s perturbed test
function approach.

Where rational normals are concerned, we only have to check that conditions
(b) and (c) are satisfied. As we will see, this is follows mmediately from the
assumptions on the operator $F$.

The same analysis also applies to Theorem 1, except that we still need to
prove that $\overline{F}^{*}=\overline{F}_{*}$. This will be done in the next
section. Once this is proved, the comparison principle implies that
$\bar{u}=\bar{u}^{*}=\bar{u}_{*}$.

The preceding discussion leads to the following result:

######  Proposition 8.

If $(u^{\epsilon})_{\epsilon>0}$ are the functions in Theorem 1 or 2 and
$\bar{u}^{*}=\limsup^{*}u^{\epsilon}$ and
$\bar{u}_{*}=\liminf_{*}u^{\epsilon}$ are the half-relaxed limits defined by

$\displaystyle\bar{u}^{*}(x,t)$ $\displaystyle=\lim_{\delta\to
0^{+}}\sup\left\\{u^{\epsilon}(y,s)\,\mid\,\epsilon+\|y-x\|+|s-t|\leq\delta\right\\},$
$\displaystyle\bar{u}_{*}(x,t)$ $\displaystyle=\lim_{\delta\to
0^{+}}\inf\left\\{u^{\epsilon}(y,s)\,\mid\,\epsilon+\|y-x\|+|s-t|\leq\delta\right\\},$

then $\bar{u}^{*}$ is a subsolution of (43) in irrational directions in
$\mathbb{R}^{d}\times(0,\infty)$ and $\bar{u}_{*}$ is a supersolution of (43)
in irrational directions in $\mathbb{R}^{d}\times(0,\infty)$.

###### Proof.

We break the proof down into three steps, one corresponding to each condition
in Definition 1. Since the argument for $\bar{u}^{*}$ is analogous, we
restrict attention to $\bar{u}_{*}$.

Assume that $\varphi:\mathbb{R}^{d}\times(0,\infty)\to\mathbb{R}$ is a smooth
function and $(x_{0},t_{0})\in\mathbb{R}^{d}\times(0,\infty)$ is a point where
$\bar{u}_{*}-\varphi$ attains a strict local minimum.

Step 1: Irrational Normal

Assume that $D\varphi(x_{0},t_{0})\notin\mathbb{R}\mathbb{Z}^{d}$. Let us
argue by contradiction. Assume that there is a $\nu>0$ such that

(44)
$\varphi_{t}(x_{0},t_{0})-\overline{F}(D\varphi(x_{0},t_{0}),D^{2}\varphi(x_{0},t_{0}))\leq-2\nu\|D\varphi(x_{0},t_{0})\|.$

Define $e=\widehat{D\varphi}(x_{0},t_{0})$ and
$X=\|D\varphi(x_{0},t_{0})\|^{-1}D^{2}\varphi(x_{0},t_{0})$. By Theorem 7, we
can let $V^{\nu}\in C^{2}(\mathbb{T}^{d})$ be an approximate corrector
satisfying

$-\nu\leq\overline{F}(e,X)-F(e,\tilde{X}_{e}+D^{2}_{e}V^{\nu},y)\leq\nu\quad\text{in}\,\,\mathbb{T}^{d}.$

For each $\epsilon>0$, define $\varphi_{\epsilon}$ by

$\varphi_{\epsilon}(x,t)=\varphi(x,t)+\epsilon^{2}\|D\varphi(x_{0},t_{0})\|V^{\nu}(\epsilon^{-1}x).$

Let $(x_{\epsilon},t_{\epsilon})$ be a local minimum of
$u^{\epsilon}-\varphi_{\epsilon}$ close to $(x_{0},t_{0})$. Since
$(x_{0},t_{0})$ is a strict minimum, we can fix a sequence
$(\epsilon_{n})_{n\in\mathbb{N}}$ such that

$\lim_{n\to\infty}\epsilon_{n}=0,\quad\lim_{n\to\infty}(x_{\epsilon_{n}},t_{\epsilon_{n}})=(x_{0},t_{0}),\quad\inf\left\\{\|D\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}})\|\,\mid\,n\in\mathbb{N}\right\\}>0.$

Invoking the equation satisfied by $u^{\epsilon_{n}}$ and using the continuity
of $F$, we find, for sufficiently large $n$,

$\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})-F(D\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}}),D^{2}\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}})+\|D\varphi(x_{0},t_{0})\|D^{2}V^{\nu}(\epsilon_{n}^{-1}x_{\epsilon_{n}}),\epsilon_{n}^{-1}x_{\epsilon_{n}})\geq
o(1).$

The continuity and geometric properties of $F$ then yield

$\displaystyle 0$
$\displaystyle\leq\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})-F(D\varphi(x_{0},t_{0}),D^{2}_{e}\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}})+\|D\varphi(x_{0},t_{0})\|D^{2}_{e}V^{\nu}(\epsilon_{n}^{-1}x_{\epsilon_{n}}),\epsilon_{n}^{-1}x_{\epsilon_{n}})+o(1)$
$\displaystyle\leq\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})-\|D\varphi(x_{0},t_{0})\|F(e,\tilde{X}_{e}+D^{2}_{e}V^{\nu}(\epsilon_{n}^{-1}x_{\epsilon_{n}}),\epsilon_{n}^{-1}x_{\epsilon_{n}})+o(1)$
$\displaystyle\leq\varphi_{t}(x_{0},t_{0})-\|D\varphi(x_{0},t_{0})\|\overline{F}(e,\tilde{X}_{e})+\nu\|D\varphi(x_{0},t_{0})\|+o(1).$

Sending $\epsilon\to 0^{+}$ and recalling the definition of $\overline{F}$ in
(34), we contradict (44). Since $\varphi$ was arbitrary, we conclude that
$\bar{u}_{*}$ satisfies condition (a) of Definition 1.

Step 2: Rational Normal, Flat Level Set

Next, we assume that $D\varphi(x_{0},t_{0})\neq 0$ and define $\delta$ by

$\delta=\|D\varphi(x_{0},t_{0})\|^{-1}\left\|\left(\text{Id}-\widehat{D\varphi}(x_{0},t_{0})\otimes\widehat{D\varphi}(x_{0},t_{0})\right)D^{2}\varphi(x_{0},t_{0})\right\|.$

Using the equation directly and invoking assumptions (6) and (7) or (3) and
(8), we find $\epsilon_{n}\to 0^{+}$ and
$(x_{\epsilon_{n}},t_{\epsilon_{n}})\to(x_{0},t_{0})$ so that

$\displaystyle 0$
$\displaystyle\leq\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})-F(D\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}}),D^{2}\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}}),\epsilon^{-1}_{n}x_{\epsilon_{n}}),$
$\displaystyle\leq\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})+\Lambda\|D^{2}_{e_{n}}\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}})\|,$

where $e_{n}=\widehat{D\varphi}(x_{\epsilon_{n}},t_{\epsilon_{n}})$. Sending
$n\to\infty$, we recover

$\varphi_{t}(x_{0},t_{0})\geq-\Lambda\|D^{2}_{e}\varphi(x_{0},t_{0})\|\geq-\Lambda\delta\|D\varphi(x_{0},t_{0})\|.$

We conclude that $\bar{u}_{*}$ satisfies condition (b) in Definition 1 with
$K=\Lambda$.

Step 3: Vanishing Normal

Finally, if $\|D\varphi(x_{0},t_{0})\|=\|D^{2}\varphi(x_{0},t_{0})\|=0$, then
we can find $\epsilon_{n}\to 0^{+}$ and
$(x_{\epsilon_{n}},t_{\epsilon_{n}})\to(x_{0},t_{0})$ such that

$\displaystyle 0$
$\displaystyle\leq\varphi_{t}(x_{\epsilon_{n}},t_{\epsilon_{n}})-F_{*}(D\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}}),D^{2}\varphi(x_{\epsilon_{n}},t_{\epsilon_{n}}),\epsilon_{n}^{-1}x_{n}).$

In the limit $n\to\infty$, we use (7) to recover $\varphi_{t}(x_{0},t_{0})\geq
0.$ This proves $\bar{u}_{*}$ satisfies condition (c) in Definition 1.∎

##  6\. Continuity and Discontinuity of Homogenized Coefficients

In this section, we study the continuity properties of the homogenized
operator $\overline{F}$ of Theorem 1 and the effective coefficients
$\overline{m}$ and $\bar{a}$ of Theorem 2. The main results are Theorem 4 and
Corollary 1 concerning the (disc)continuity properties of $\overline{m}$ and
$\overline{a}$ in the quasilinear setting.

In dimension two, it turns out that the homogenized operator obtained in
Theorem 1 is always continuous, a result that is stated next.

######  Proposition 9.

Under the assumptions of Theorem 1, the homogenized operator
$\overline{F}:(\mathbb{R}^{2}\setminus\mathbb{R}\mathbb{Z}^{2})\times\mathcal{S}_{2}\to\mathbb{R}$
(see (34)) extends continuously to
$(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}$.

For the sake of completeness, we show how Proposition 9 completes the proof of
Theorem 1:

###### Proof of Theorem 1.

Let $\bar{u}^{*}$ and $\bar{u}_{*}$ be the half-relaxed limits defined in
Proposition 8. That proposition shows that $\bar{u}^{*}$ and $\bar{u}_{*}$ are
respectively sub- and supersolution of (43) in
$\mathbb{R}^{d}\times(0,\infty)$. Furthermore, using (7) and the assumption
$u_{0}\in UC(\mathbb{R}^{d})$, we can build sub- and supersolutions that show
$\bar{u}^{*}=\bar{u}_{*}=u_{0}$ on $\mathbb{R}^{d}\times\\{0\\}$. Since
$\overline{F}^{*}=\overline{F}_{*}$ in
$(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}$, the comparison
principle implies $\bar{u}^{*}\leq\bar{u}_{*}$. Therefore,
$\bar{u}^{*}=\bar{u}_{*}=\bar{u}$ and we conclude that
$u^{\epsilon}\to\bar{u}$ locally uniformly in
$\mathbb{R}^{d}\times[0,\infty)$. ∎

###  6.1. Strategy of Proof

Before entering into the details, let us briefly give a heuristic explanation
of the strategy of the proof and the core technical issues that arise. The
main idea of the proof and the discussion that follows are inspired by [27].

Suppose $e\in S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$. In the proof of Theorem 4,
we proceed by studying the behavior of $(\bar{\mu}_{e_{n}})_{n\in\mathbb{N}}$
along a sequence of irrational directions $(e_{n})_{n\in\mathbb{N}}$ with
$e_{n}\to e$ and $\frac{e_{n}-e}{\|e_{n}-e\|}\to-\eta$.

$\bar{\mu}_{e_{n}}$ captures the behavior of the diffusion $X^{e_{n}}$ of (15)
after long times. Therefore, it is natural to pass to the diffusions
$X^{e_{n}}$ and consider their behavior. Further, in light of the structure of
$\mathscr{I}^{a}_{e}$, the only question is the $e$ marginal of any limit
point of $(\bar{\mu}_{e_{n}})_{n\in\mathbb{N}}$, that is, we only need to
study $f(\langle X^{e_{n}}_{t},e\rangle)$, where $f$ is an $r_{e}$-periodic
function of one variable and $t>0$ is large.

We may as well assume that $\langle X^{e_{n}}_{0},e_{n}\rangle=0$, which
implies that $\langle X^{e_{n}}_{t},e_{n}\rangle=0$ for all $t\geq 0$.
Therefore, if we write $e_{n}=\cos(\theta_{n})e-\sin(\theta_{n})\eta_{n}$ for
some $\theta_{n}\in(-\pi,\pi]$ and $\eta_{n}\in S^{d-1}\cap\langle
e\rangle^{\perp}$, we have

$\langle X^{e_{n}}_{t},e\rangle=\langle
X^{e_{n}}_{t},e-e_{n}\rangle=(1-\cos(\theta_{n})\langle
X^{e_{n}}_{t},e\rangle+O(\theta_{n}^{2}).$

Hence to recover anything meaningful from $f(\langle X^{e_{n}}_{t},e\rangle)$,
we need to wait until $\|X^{e_{n}}_{t}\|\approx\theta_{n}^{-1}$. Since this
takes a time proportional to $\theta_{n}^{-2}$, we should study $f(\langle
X^{e_{n}}_{\theta_{n}^{-2}T},e\rangle)$ in the simultaneous limit
$n,T\to\infty$.

Put another way, at the level of the PDE, we would like to understand the
behavior as $n\to\infty$ and $\delta\to 0^{+}$ of the penalized correctors
$(V^{\delta}_{n})_{(\delta,n)\in(0,\infty)\times\mathbb{N}}$ solving

$\delta
V^{\delta}_{n}-\text{tr}\left(a(\theta_{n}^{-1}y,e_{n})D^{2}_{e}V^{\delta}_{n}\right)=f(\theta_{n}^{-1}\langle
x,e\rangle)\quad\text{in}\,\,\langle e_{n}\rangle^{\perp}.$

A simple homogenization argument shows that, for a fixed $\delta>0$,
$V^{\delta}_{n}\to\tilde{V}^{\delta}$ as $n\to\infty$, where
$\tilde{V}^{\delta}$ is the solution of the problem

$\delta\tilde{V}^{\delta}-\text{tr}\left(a^{\perp}_{e}(\langle\eta,x\rangle
e)D^{2}_{e}\tilde{V}^{\delta}\right)=f(\langle\eta,x\rangle
e)\quad\text{in}\,\,\langle e\rangle^{\perp}.$

It turns out that extracting the limit of $\delta\tilde{V}^{\delta}$ as
$\delta\to 0^{+}$ leads to the correct limit of
$\int_{\mathbb{T}^{d}}f(\langle y,e\rangle)\,\bar{\mu}_{e_{n}}(dy)$. The
question is simply how to show that the limits $n\to\infty$ and $\delta\to
0^{+}$ commute.

The argument below shows how to do this working at the level of the obstacle
problems introduced in [16] rather than the penalized correctors. Working with
penalized correctors is difficult since it requires understanding the rate at
which $\delta V^{\delta}_{n}$ converges as $\delta\to 0^{+}$, independently of
$n$. We circumvent this by passing to the obstacle problem approach of [16]
and leveraging an upper semi-continuity property proved there. In this way, it
is possible to first send $n\to\infty$ and then $\delta\to 0^{+}$ without
explicitly quantifying the rates of convergence of the associated almost
periodic homogenization problems.

###  6.2. Preliminaries

We start by introducing some notation that will be used later.

Henceforth let $F$ be the operator in (1) satisfying either the assumptions of
Theorem 1 or Theorem 2 with
$F(p,X,y)=m(y,\hat{p})^{-1}\text{tr}(a(y,\hat{p})\tilde{X}_{\hat{p}})$. It
will be convenient to define the family of shifted operators
$\\{F^{x}\\}_{x\in\mathbb{T}^{d}}$ by

$F^{x}(p,X,y)=F(p,X,y+x).$

The reader familiar with stochastic homogenization can think of the shift $x$
like an element $\omega$ of a probability space $(\Omega,\mathbb{P})$. In our
case, $\Omega=\mathbb{T}^{d}$ and $\mathbb{P}=\mathcal{L}^{d}$.

To prove Proposition 9 and Theorem 4, we will begin with an $e\in
S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$, an $\eta\in S^{d-1}\cap\langle
e\rangle^{\perp}$, and a sequence $(e_{n})_{n\in\mathbb{N}}\subseteq
S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$ such that

$\lim_{n\to\infty}\left(\|e_{n}-e\|+\left\|\frac{e_{n}-e}{\|e_{n}-e\|}+\eta\right\|\right)=0.$

We have to prove that the sequences
$(\overline{F}(e_{n},\cdot))_{n\in\mathbb{N}}$ or
$(\bar{\mu}_{e_{n}})_{n\in\mathbb{N}}$ have the desired behavior in the limit.
For the rest of the section, $e$, $\eta$, and $(e_{n})_{n\in\mathbb{N}}$ are
fixed.

In view of the preceding, we can fix $(\eta_{n})_{n\in\mathbb{N}}\subseteq
S^{d-1}$ and $(\theta_{n})_{n\in\mathbb{N}}\subseteq(-\pi,\pi]$ such that

$e_{n}=\cos(\theta_{n})e-\sin(\theta_{n})\eta_{n}.$

We will assume without loss of generality that
$(\theta_{n})_{n\in\mathbb{N}}\subseteq(-\pi,\pi)\setminus\\{0\\}$.

For each $n\in\mathbb{N}$, let $O_{n}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the
orthogonal transformation such that

$O_{n}(e_{n})=e,\quad
O_{n}(\sin(\theta_{n})e+\cos(\theta_{n})\eta_{n})=\eta_{n},\quad
O_{n}\restriction_{\langle
e,e_{n}\rangle^{\perp}}=\text{Id}\restriction_{\langle
e,e_{n}\rangle^{\perp}}$

Notice that $\lim_{n\to\infty}O_{n}=\text{Id}$ and $O_{n}(\langle
e_{n}\rangle^{\perp})=\langle e\rangle^{\perp}$.

Finally, we let $\\{v_{1},\dots,v_{d-1}\\}$ be an orthonormal basis of
$\langle e\rangle^{\perp}$ and define, for each $R>0$, the cube
$Q^{*}_{R}\subseteq\langle e\rangle^{\perp}$ by

$Q^{*}_{R}=\left\\{y\in\langle e\rangle^{\perp}\,\mid\,|\langle
y,v_{1}\rangle|\vee\dots\vee|\langle y,v_{d-1}\rangle|<R/2\right\\}.$

Given $n\in\mathbb{N}$, we define the analogous cubes
$\\{Q^{(n)}_{R}\,\mid\,R>0\\}$ in $\langle e_{n}\rangle^{\perp}$ by
$Q^{(n)}_{R}=O_{n}^{-1}(Q^{*}_{R})$.

###  6.3. Obstacle Problems

It is technically very convenient to replace the penalized correctors
$(V^{\delta}_{n})_{(\delta,n)}$ of the previous discussion by solutions of a
related family of obstacle problems. In this section, we describe the set-up.

Given $n\in\mathbb{N}$, $\mu\in\mathbb{R}$, $R>0$, $X\in\mathcal{S}_{d}$, and
$x\in\mathbb{T}^{d}$, we set $\nu=(n,\mu,R,X)$ and define the obstacle sub-
and supersolutions $u^{\nu,x}$ and $u_{\nu,x}$ as the solutions of the
equations

(47) $\displaystyle\left\\{\begin{array}[]{r
l}\max\left\\{-F^{x}(e_{n},\tilde{X}_{e_{n}}+D^{2}_{e_{n}}u^{\nu,x},\theta_{n}^{-1}y)-\mu,u^{\nu,x}\right\\}=0&\text{in}\,\,Q^{(n)}_{R},\\\
u^{\nu,x}=0&\text{on}\,\,\partial Q^{(n)}_{R},\end{array}\right.$ (50)
$\displaystyle\left\\{\begin{array}[]{r
l}\min\left\\{-F^{x}(e_{n},\tilde{X}_{e_{n}}+D^{2}_{e_{n}}u_{\nu,x},\theta_{n}^{-1}y)-\mu,u_{\nu,x}\right\\}=0&\text{in}\,\,Q^{(n)}_{R},\\\
u_{\nu,x}=0&\text{on}\,\,\partial Q^{(n)}_{R}.\end{array}\right.$

Existence and uniqueness for these problems can be proved using a comparison
principle or through a penalization procedure as in [16].

When $\nu=(*,\mu,R,X)$, $u^{\nu,x}$ and $u_{\nu,x}$ are the solutions of the
homogenized problems

(53) $\displaystyle\left\\{\begin{array}[]{r
l}\max\left\\{-F_{e}^{\perp}(\tilde{X}_{e}+D^{2}_{e}u^{\nu,x},\langle
y,\eta\rangle e+\langle x,e\rangle
e)-\mu,u^{\nu,x}\right\\}=0&\text{in}\,\,Q^{*}_{R},\\\
u^{\nu,x}=0&\text{on}\,\,\partial Q^{*}_{R},\end{array}\right.$ (56)
$\displaystyle\left\\{\begin{array}[]{r
l}\min\left\\{-F_{e}^{\perp}(\tilde{X}_{e}+D^{2}_{e}u^{\nu,x},\langle
y,\eta\rangle e+\langle x,e\rangle
e)-\mu,u^{\nu,x}\right\\}=0&\text{in}\,\,Q^{*}_{R},\\\
u^{\nu,x}=0&\text{on}\,\,\partial Q^{*}_{R}.\end{array}\right.$

For these equations, well-posedness is clear in the context of Theorem 1 since
$\langle e\rangle^{\perp}$ is one-dimensional in this case. Where Theorem 2 is
concerned, the regularity of $a^{\perp}_{e}$ and $m^{\perp}_{e}$ proved in
Proposition 7 is more than enough to guarantee well-posedness.

Next, for $n\in\mathbb{N}\cup\\{*\\}$, we define the contact sets $K^{\nu,x}$
and $K_{\nu,x}$ by

$K^{\nu,x}=\\{y\in Q^{(n)}_{R}\,\mid\,u^{\nu,x}(y)=0\\},\quad
K_{\nu,x}=\\{y\in Q^{(n)}_{R}\,\mid\,u_{\nu,x}(y)=0\\}.$

As shown in [16], for each $n\in\mathbb{N}\cup\\{*\\}$, there are functions
$\overline{\ell}_{n},\underline{\ell}_{n}:\mathbb{R}\times\mathcal{S}_{d}\to[0,1]$
such that, for each $(\mu,X)\in\mathbb{R}\times\mathcal{S}_{d}$, for
$\nu(R)=(n,\mu,R,X)$, we have

$\displaystyle\overline{\ell}_{n}(\mu,X)=\lim_{R\to\infty}R^{1-d}\mathcal{H}^{d-1}(K^{\nu(R),x})\quad\text{for
a.e.}\,\,x\in\mathbb{T}^{d},$
$\displaystyle\underline{\ell}_{n}(\mu,X)=\lim_{R\to\infty}R^{1-d}\mathcal{H}^{d-1}(K_{\nu(R),x})\quad\text{for
a.e.}\,\,x\in\mathbb{T}^{d}.$

Furthermore, these functions satisfy the following variational principles:

(57) $\displaystyle\overline{\ell}_{n}(\mu,X)$
$\displaystyle=\inf\left\\{R^{1-d}\int_{\mathbb{T}^{d}}\mathcal{H}^{d-1}(K^{\nu(R),x})\,dx\,\mid\,R>0\right\\},$
$\displaystyle\underline{\ell}_{n}(\mu,X)$
$\displaystyle=\inf\left\\{R^{1-d}\int_{\mathbb{T}^{d}}\mathcal{H}^{d-1}(K_{\nu(R),x})\,dx\,\mid\,R>0\right\\}.$

###  6.4. Homogenization of Obstacle Problems

By a direct analogy to the discussion of the penalized correctors above, we
now show that the obstacle problems (47) and (50) homogenize to the problems
(53) and (56) as $n\to\infty$.

######  Proposition 10.

Given $(\mu,X,R)>0$ and $x\in\mathbb{T}^{d}$, if $\nu_{n}=(n,\mu,X,R)$ and
$\nu_{*}=(*,\mu,X,R)$, then

$\displaystyle 0=\lim_{\delta\to
0^{+}}\sup\left\\{|u^{\nu_{n},x}(y_{1})-u^{\nu_{*},x}(y_{2})|\,\mid\,(y_{1},y_{2})\in
Q^{(n)}_{R}\times Q^{*}_{R},\,\,n^{-1}+\|y_{1}-y_{2}\|<\delta\right\\},$
$\displaystyle 0=\lim_{\delta\to
0^{+}}\sup\left\\{|u_{\nu_{n},x}(y_{1})-u_{\nu_{*},x}(y_{2})|\,\mid\,(y_{1},y_{2})\in
Q^{(n)}_{R}\times Q^{*}_{R},\,\,n^{-1}+\|y_{1}-y_{2}\|<\delta\right\\}.$

###### Proof.

To prove this, we will work with half-relaxed limits. The proof for
supersolutions follows by analogous arguments so we will restrict attention to
subsolutions.

Define $\bar{u}^{*}$ and $\bar{u}_{*}$ in $\overline{Q^{*}_{R}}$ by

$\displaystyle\bar{u}^{*}(y)$ $\displaystyle=\lim_{\delta\to
0^{+}}\sup\left\\{u^{\nu_{n},x}(y^{\prime})\,\mid\,y^{\prime}\in
Q^{(n)}_{R},\,\,n^{-1}+\|y^{\prime}-y\|<\delta\right\\},$
$\displaystyle\bar{u}_{*}(y)$ $\displaystyle=\lim_{\delta\to
0^{+}}\inf\left\\{u^{\nu_{n},x}(y^{\prime})\,\mid\,y^{\prime}\in
Q^{(n)}_{R},\,\,n^{-1}+\|y^{\prime}-y\|<\delta\right\\}.$

It suffices to show that $\bar{u}^{*}=\bar{u}_{*}=u^{\nu_{*},x}$. To do this,
we only need to prove that $\bar{u}^{*}$ and $\bar{u}_{*}$ are sub- and
supersolution of (53) and apply the comparison principle. Since the proof for
$\bar{u}^{*}$ is almost identical, we will restrict attention to
$\bar{u}_{*}$.

Assume that $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ is a smooth function and
$y_{0}\in Q^{*}_{R}$ is a point where $\bar{u}_{*}-\varphi$ has a strict local
minimum. We claim that

(58)
$\max\left\\{-F^{\perp}_{e}(\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0}),-\langle
y_{0},\eta\rangle e+\langle x,e\rangle e)-\mu,\bar{u}_{*}(y_{0})\right\\}\geq
0.$

Clearly, we can assume that $\bar{u}_{*}(y_{0})<0$.

Let us argue by contradiction. If (58) fails to hold, then there is a $\nu>0$
such that

(59) $-F^{\perp}_{e}(\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0}),\langle
y_{0},\eta\rangle e+\langle x,e\rangle e)-\mu<-\nu.$

Let $V\in C^{2}(\mathbb{T}^{d})$ be an approximate corrector satisfying the
equation

$-\nu\leq F^{\perp}_{e}(\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0}),\langle
y,e\rangle e+\langle x,e\rangle
e)-F^{x}(e,\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0})+D^{2}_{e}V,y)\leq\nu\quad\text{in}\,\,\mathbb{T}^{d}.$

For each $n\in\mathbb{N}$, let $y_{n}\in\overline{Q^{(n)}_{R}}$ be a point
where the function $y\mapsto
u^{\nu_{n},x}(y)-\varphi(y)-\theta_{n}^{2}V(\theta_{n}^{-1}y)$ attains its
minimum. By classical arguments, we can pass to a subsequence
$(y_{n_{j}})_{j\in\mathbb{N}}$ such that

$\lim_{j\to\infty}y_{n_{j}}=y_{0},\quad\lim_{j\to\infty}u^{\nu_{n_{j}},x}(y_{n_{j}})=\bar{u}_{*}(y_{0})<0.$

There is no loss of generality in assuming that $y_{n_{j}}\in Q^{(n_{j})}_{R}$
for all $j\in\mathbb{N}$ and $u^{\nu_{n_{j}},x}(y_{n_{j}})<0$ for all
$j\in\mathbb{N}$. Thus, the equation satisfied by $u^{\nu_{n_{j}},x}$ gives

$\displaystyle 0$
$\displaystyle\leq-F^{x}(e_{n_{j}},\tilde{X}_{e_{n_{j}}}+D^{2}_{e_{n_{j}}}\varphi(y_{n_{j}})+D^{2}_{e_{n_{j}}}V(\theta_{n_{j}}^{-1}y_{n_{j}}),\theta_{n_{j}}^{-1}y_{n_{j}})-\mu$
$\displaystyle=-F^{x}(e,\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0})+D^{2}_{e}V(\theta_{n_{j}}^{-1}y_{n_{j}}),\theta_{n_{j}}^{-1}y_{n_{j}})-\mu+o(1)$
$\displaystyle\leq-F^{\perp}_{e}(\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0}),\theta_{n_{j}}^{-1}\langle
y_{n_{j}},e\rangle e+\langle x,e\rangle e)-\mu+\nu+o(1).$

Since $y_{n_{j}}\in\langle e_{n_{j}}\rangle^{\perp}$, we have

$\displaystyle\theta_{n_{j}}^{-1}\langle y_{n_{j}},e\rangle$
$\displaystyle=\theta_{n_{j}}^{-1}\langle
y_{n_{j}},e-e_{n_{j}}\rangle=\left(\frac{1-\cos(\theta_{n_{j}})}{\theta_{n_{j}}}\right)\langle
e,y_{n_{j}}\rangle+\left(\frac{\sin(\theta_{n_{j}})}{\theta_{n_{j}}}\right)\langle\eta_{n_{j}},y_{n_{j}}\rangle.$

Thus, $\theta_{n_{j}}^{-1}\langle y_{n_{j}},e\rangle\to\langle
y_{0},\eta\rangle$ and we find, in the limit $n\to\infty$,

$-\nu\leq-F^{\perp}_{e}(\tilde{X}_{e}+D^{2}_{e}\varphi(y_{0}),\langle
y_{0},\eta\rangle e+\langle x,e\rangle e)-\mu.$

This directly contradicts (59).

We deduce that $\bar{u}_{*}$ is a supersolution of (53) in the interior of
$Q^{*}_{R}$. Using barriers, it is not hard to show that $\bar{u}_{*}\geq 0$
in $\partial Q^{*}_{R}$. Thus, $\bar{u}_{*}$ is a supersolution. ∎

###  6.5. Densities of Contact Sets

Once we state some properties of the densities $\overline{\ell}_{n}$ and
$\underline{\ell}_{n}$, we will have all the tools necessary to prove
Proposition 9 and Theorem 4.

The following result follows by arguing as in [16] (also see [2]). In the
statement, we write $e_{*}=e$.

######  Proposition 11.

For each $X\in\mathcal{S}_{d}$, there is a sequence
$(\mu_{n}(X))_{n\in\mathbb{N}\cup\\{*\\}}\subseteq\mathbb{R}$ such that, for
each $n\in\mathbb{N}\cup\\{*\\}$, the following statements hold:

  * (i)

$\overline{\ell}_{n}(\mu,X)=0$ if $\mu<\mu_{n}(X)$ and
$\overline{\ell}_{n}(\mu,X)\geq c|\mu-\mu_{n}|^{d}$ if $\mu>\mu_{n}(X)$.

  * (ii)

$\underline{\ell}_{n}(\mu,X)=0$ if $\mu>\mu_{n}(X)$ and
$\underline{\ell}_{n}(\mu,X)\geq c|\mu-\mu_{n}|^{d}$ if $\mu<\mu_{n}(X)$.

  * (iii)

If $x\in\mathbb{T}^{d}$, $n\in\mathbb{N}\cup\\{*\\}$, and, for each
$\nu=(n,\mu,X,R)$, $v^{\nu,x}$ is the solution of the Dirichlet problem

$\left\\{\begin{array}[]{r
l}-F^{x}(e_{n},\tilde{X}_{e_{n}}+D^{2}_{e_{n}}v^{\nu,x},y)=\mu_{n}(X)&\text{in}\,\,Q^{(n)}_{R},\\\
v^{\nu,x}=0&\text{on}\,\,\partial Q^{(n)}_{R},\end{array}\right.$

then

$\lim_{R\to\infty}\sup\left\\{R^{-2}|v^{\nu(R),x}(y)|\,\mid\,y\in
Q^{(n)}_{R}\right\\}=0.$

(The constant $c>0$ depends on $\lambda$, $\Lambda$, and $d$, but not on $n$.)

In addition, we will need the following fact, adapted from [16], that follows
from the homogenization result above:

######  Proposition 12.

For each $\mu\in\mathbb{R}$ and $X\in\mathcal{S}_{d}$, we have

$\overline{\ell}_{*}(\mu,X)\geq\limsup_{n\to\infty}\overline{\ell}_{n}(\mu,X),\quad\underline{\ell}_{*}(\mu,X)\geq\limsup_{n\to\infty}\underline{\ell}_{n}(\mu,X).$

###### Proof.

We give the details for $(\overline{\ell}_{n})_{n\in\mathbb{N}}$; the same
basic idea also applies to $(\underline{\ell}_{n})_{n\in\mathbb{N}}$.

Step 1: Property of Contact Sets

Fix $(\mu,X,R)\in\mathbb{R}\times\mathcal{S}_{d}\times(0,\infty)$ and
$x\in\mathbb{T}^{d}$ and let $\nu_{n}=(n,\mu,X,R)$ and $\nu_{*}=(*,\mu,X,R)$.
We claim that

(60)
$\mathcal{H}^{d-1}(K^{\nu_{*},x})\geq\limsup_{n\to\infty}\mathcal{H}^{d-1}(K^{\nu_{n},x}).$

To see this, first, define $\tilde{K}\subseteq Q^{*}_{R}$ by

$\tilde{K}=\bigcap_{n=1}^{\infty}\bigcup_{m=n}^{\infty}\left\\{y\in
Q^{*}_{R}\,\mid\,u^{\nu_{m},x}(O_{m}^{-1}(y))=0\right\\}.$

Notice that Proposition 10 implies that $\tilde{K}\subseteq K^{\nu_{*},x}$.
Therefore, the measure preserving property of $O_{m}$ yields

$\displaystyle\mathcal{H}^{d-1}(K^{\nu_{*},x})$
$\displaystyle\geq\mathcal{H}^{d-1}(\tilde{K})$
$\displaystyle=\lim_{n\to\infty}\mathcal{H}^{d-1}\left(\bigcup_{m=n}^{\infty}\left\\{y\in
Q^{*}_{R}\,\mid\,u^{\nu_{m},x}(O_{m}^{-1}(y))=0\right\\}\right)$
$\displaystyle\geq\limsup_{n\to\infty}\mathcal{H}^{d-1}(K^{\nu_{n},x}).$

Step 2: Conclusion

Finally, given $\epsilon>0$, we invoke (57) to fix an $R>0$ such that

$\overline{\ell}_{*}(\mu,X)\geq
R^{1-d}\int_{\mathbb{T}^{d}}\mathcal{H}^{d-1}(K^{\nu_{*},x})\,dx-\epsilon.$

By (60), we can apply (57) and Fatou’s Lemma (bearing in mind the functions in
question are bounded) to find

$\displaystyle\int_{\mathbb{T}^{d}}\mathcal{H}^{d-1}(K^{\nu_{*},x})\,dx\geq\limsup_{n\to\infty}\int_{\mathbb{T}^{d}}\mathcal{H}^{d-1}(K^{\nu_{n},x})\,dx\geq\limsup_{n\to\infty}R^{d-1}\overline{\ell}_{n}(\mu,X).$

Thus,

$\overline{\ell}_{*}(\mu,X)\geq\limsup_{n\to\infty}\overline{\ell}_{n}(\mu,X)-\epsilon$

and we conclude upon sending $\epsilon\to 0^{+}$. ∎

Finally, combining Propositions 11 and 12, we obtain

######  Proposition 13.

For each $X\in\mathcal{S}_{d}$, $\mu_{*}(X)=\lim_{n\to\infty}\mu_{n}(X)$.

###### Proof.

We will show that $\mu_{*}(X)\geq\limsup_{n\to\infty}\mu_{n}(X)$. The proof
that $\mu_{*}(X)\leq\liminf_{n\to\infty}\mu_{n}(X)$ proceeds analogously,
replacing $(\underline{\ell}_{n})_{n\in\mathbb{N}}$ by
$(\overline{\ell}_{n})_{n\in\mathbb{N}}$.

Suppose that $\mu<\limsup_{n\to\infty}\mu_{n}(X)$. By Proposition 11,

$\underline{\ell}_{*}(\mu,X)\geq\limsup_{n\to\infty}\underline{\ell}_{n}(\mu,X)\geq
c\limsup_{n\to\infty}(\mu-\mu_{n})_{-}^{d}>0.$

From this, the same proposition with $n=*$ yields $\mu<\mu_{*}(X)$. Since
$\mu$ was arbitrary, we conclude
$\limsup_{n\to\infty}\mu_{n}(X)\leq\mu_{*}(X)$. ∎

###  6.6. Limiting behavior at $e$

The results of the previous section directly imply Proposition 9 and Theorem
4, as we now show.

To start with, we note that the constants
$(\mu_{n}(X))_{n\in\mathbb{N}\cup\\{*\\}}$ of Proposition 11 are precisely the
homogenized coefficients.

######  Proposition 14.

For each $n\in\mathbb{N}$, $\mu_{n}(X)=-\overline{F}(e_{n},X)$.

###### Proof.

Fix $n\in\mathbb{N}$ and $X\in\mathcal{S}_{d}$. By Proposition 11, if we let
$(v_{\epsilon})_{\epsilon>0}$ be the solutions of the Dirichlet problem

$\left\\{\begin{array}[]{r
l}-F(e_{n},\tilde{X}_{e_{n}}+D^{2}_{e_{n}}v_{\epsilon},\epsilon^{-1}x^{\prime})=\mu_{n}(X)&\text{in}\,\,Q^{(n)}_{1},\\\
v_{\epsilon}=0&\text{on}\,\,\partial Q^{(n)}_{1},\end{array}\right.$

then $v_{\epsilon}\to 0$ uniformly in $Q^{(n)}_{1}$ as $\epsilon\to 0^{+}$.

At the same time, this is an almost periodic elliptic homogenization problem
in $\langle e_{n}\rangle^{\perp}$ and we know (e.g. using the approximate
correctors of Theorem 7) that $v_{\epsilon}\to\bar{v}$, where $\bar{v}$ is the
solution of

$\left\\{\begin{array}[]{r
l}-\overline{F}(e_{n},\tilde{X}_{e_{n}}+D^{2}_{e_{n}}\bar{v})=\mu_{n}(X)&\text{in}\,\,Q^{(n)}_{1}\\\
\bar{v}=0&\text{on}\,\,\partial Q^{(n)}_{1}\end{array}\right.$

The previous paragraph says that $\bar{v}\equiv 0$ in $Q^{(n)}_{1}$.
Therefore,

$\mu_{n}(X)=-\overline{F}(e_{n},\tilde{X}_{e_{n}})=-\overline{F}(e_{n},X).$

∎

######  Proposition 15.

If $F$ satisfies the assumptions of Theorem 1, then, for each $e\in
S^{1}\cap\mathbb{R}\mathbb{Z}^{2}$ and $X\in\mathcal{S}_{2}$,

$-\mu_{*}(X)=\lim_{\mathbb{R}\mathbb{Z}^{2}\not\ni e^{\prime}\to
e}\overline{F}(e^{\prime},X).$

In particular, $\overline{F}$ extends to a continuous function in
$(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}_{2}$. Furthermore,
$X\mapsto-\mu_{*}(X)$ is the homogenized coefficient associated with the (e.g.
Dirichlet) elliptic homogenization problem:

(61) $\left\\{\begin{array}[]{r
l}U-F^{\perp}_{e}(D^{2}_{e}U,\epsilon^{-1}\langle x,e^{\perp}\rangle
e)=0&\text{in}\,\,Q^{*}_{1}\\\ U=G&\text{on}\,\,\partial
Q^{*}_{1}\end{array}\right.$

(Here $e^{\perp}$ is any unit vector with $\langle e^{\perp},e\rangle=0$.)

###### Proof.

Fix $e^{\perp}\in S^{1}\cap\langle e\rangle^{\perp}$ and note that
$S^{1}\cap\langle e\rangle^{\perp}=\\{e^{\perp},-e^{\perp}\\}$.

Assume that $(e_{n})_{n\in\mathbb{N}}\subseteq
S^{1}\setminus\mathbb{R}\mathbb{Z}^{2}$ and $e_{n}\to e$. Passing to a
subsequence if necessary, we can assume there is a $u\in\\{-1,1\\}$ such that
$\lim_{n\to\infty}\|e_{n}-e\|^{-1}(e_{n}-e)=ue^{\perp}$.

Propositions 13 and 14 shows that there is a constant $\mu_{*}(X)$ such that
$\overline{F}(e_{n},X)\to-\mu_{*}(X)$. Arguing as in Proposition 14 and
replacing $e^{\perp}$ by $-e^{\perp}$ if necessary in (61), we see that
$-\mu_{*}(X)$ is the homogenized coefficient associated with (61). At the same
time, a direct manipulation of (61) shows this coefficient is unchanged if
$e^{\perp}$ is replaced by $-e^{\perp}$. Therefore, $-\mu_{*}(X)$ does not
depend on $u$ or the sequence $(e_{n})_{n\in\mathbb{N}}$ and the result
follows. ∎

The quasi-linear problem can be treated in the same way. However, as soon as
$d\geq 3$, the limiting coefficient $\mu_{*}$ has a non-trivial dependence on
the direction $\eta$.

######  Proposition 16.

If $m$ and $a$ satisfy the assumptions of Theorem 2 and
$F(p,X,y)=m(y,\hat{p})^{-1}\text{tr}(a(y,\hat{p})\tilde{X}_{\hat{p}})$, then

(62)
$-\mu_{*}(X)=(\tilde{m}_{e}^{\eta})^{-1}\text{tr}(\tilde{a}^{\eta}_{e}\tilde{X}_{e}),$

where $\tilde{a}^{\eta}_{e}$ and $\tilde{m}^{\eta}_{e}$ are given by

$\tilde{a}^{\eta}_{e}=\int_{\mathbb{T}^{d}}a(y,e)\,\tilde{\mu}_{e}^{\eta}(dy),\quad\tilde{m}^{\eta}_{e}=\int_{\mathbb{T}^{d}}m(y,e)\,\tilde{\mu}_{e}^{\eta}(dy).$

(Here $\tilde{\mu}^{\eta}_{e}$ is the measure defined in Theorem 4.)
Therefore,

$\lim_{\delta\to
0^{+}}\sup\left\\{\|\bar{a}(e^{\prime})-\tilde{a}^{\eta}_{e}\|+|\overline{m}(e^{\prime})-\tilde{m}^{\eta}_{e}|\,\mid\,\|e^{\prime}-e\|+\left\|\frac{e^{\prime}-e}{\|e^{\prime}-e\|}-\eta\right\|<\delta\right\\}=0.$

###### Proof.

As in Proposition 15, $-\mu_{*}(X)$ is the homogenized coefficient associated
with the homogenization problem

$\left\\{\begin{array}[]{r l}m^{\perp}_{e}(\epsilon^{-1}\langle
x^{\prime},\eta\rangle e)-\text{tr}\left(a^{\perp}_{e}(\epsilon^{-1}\langle
x^{\prime},\eta\rangle e)D^{2}_{e}U\right)=0&\text{in}\,\,Q^{*}_{1},\\\
U=G&\text{on}\,\,\partial Q^{*}_{1}.\end{array}\right.$

Since the coefficients only vary in the $\eta$ direction, the homogenized
coefficients are the same as if we consider the one-dimensional problem. In
particular, by a well-known computation, $-\mu_{*}(X)$ is given by (62), where
$\tilde{a}^{\eta}_{e}$ and $\tilde{m}^{\eta}_{e}$ are defined by

$\displaystyle\tilde{a}^{\eta}_{e}$
$\displaystyle=\left(\int_{0}^{r_{e}}\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds\right)^{-1}\int_{0}^{r_{e}}a_{e}^{\perp}(se)\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds,$
$\displaystyle\tilde{m}^{\eta}_{e}$
$\displaystyle=\left(\int_{0}^{r_{e}}\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds\right)^{-1}\int_{0}^{r_{e}}m_{e}^{\perp}(se)\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds.$

We conclude by recalling the definitions of $a^{\perp}_{e}$ and
$m^{\perp}_{e}$ (see (19) and (40)).

The final statement follows from the fact that $(e_{n})_{n\in\mathbb{N}}$ was
an arbitrary sequence in $S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$ with
$e_{n}\to e$ and $\frac{e_{n}-e}{\|e_{n}-e\|}\to-\eta$. ∎

###  6.7. Generic Discontinuities

In light of the formulas obtained for the limiting measures in the previous
section, it is natural to expect that $\bar{a}$ and $\bar{m}$ are generically
discontinuous at some rational directions. In this section, we prove that, in
fact, $\bar{a}$ and $\bar{m}$ are generically discontinuous at every rational
direction when $d\geq 3$.

###### Proof of Corollary 1.

To start with, since $d\geq 3$, we can fix $e\in S^{d-1}$ and let
$(\eta_{n})_{n\in\mathbb{N}}$ be a sequence of points in $S^{d-1}\cap\langle
e\rangle^{\perp}$ with $\eta_{n}\notin\\{\eta_{m},-\eta_{m}\\}$ for all $n\neq
m$. The goal is to prove that, for each $n,m\in\mathbb{N}$ with $n\neq m$, the
following sets are open and dense in
$C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))$ in the
$C^{2,\alpha}$ norm topology:

$\displaystyle\mathcal{U}_{e}(n,m)=\left\\{a\in
C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))\,\mid\,\tilde{\mu}_{e}^{\eta_{n}}\neq\tilde{\mu}_{e}^{\eta_{m}}\right\\},$
$\displaystyle\mathcal{V}_{e}(n,m)=\left\\{a\in
C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))\,\mid\,\tilde{a}_{e}^{\eta_{n}}\neq\tilde{a}_{e}^{\eta_{m}}\right\\}.$

That these sets are open is immediate. It only remains to show they are dense.

We start with $\mathcal{U}_{e}(n,m)$. It turns out that we only need to
understand the derivative of the map $a\mapsto a_{e}^{\perp}$. Toward that
end, for each $n\in\mathbb{N}$, define $u_{n}\in C^{2,\alpha}(\mathbb{T}^{d})$
to be the solution of the cell problem (41) with $f(y)=\langle
a(y)\eta_{n},\eta_{n}\rangle$. Notice that the oscillating function
$f_{e}^{\perp}$ equals $\langle a_{e}^{\perp}\eta_{n},\eta_{n}\rangle$ in this
case.

Step 1: Perturb so that $D^{2}_{e}u_{n}\neq D^{2}_{e}u_{m}$

To start with, we claim that there is no loss of generality in assuming that
$D^{2}_{e}u_{n}\neq D^{2}_{e}u_{m}$. Indeed, if
$D^{2}_{e}u_{n}=D^{2}_{e}u_{m}$, then (41) implies
$\langle[a-a^{\perp}_{e}]\eta_{n},\eta_{n}\rangle=\langle[a-a^{\perp}_{e}]\eta_{m},\eta_{m}\rangle$.
That is,

$\langle a(y)\eta_{n},\eta_{n}\rangle-\langle
a(y)\eta_{m},\eta_{m}\rangle=\langle a_{e}^{\perp}(\langle y,e\rangle
e)\eta_{n},\eta_{n}\rangle-\langle a_{e}^{\perp}(\langle y,e\rangle
e)\eta_{m},\eta_{m}\rangle\quad\text{if}\,\,y\in\mathbb{T}^{d}.$

In particular, the left-hand side varies only in the $e$ direction. This
symmetry is easily broken, for instance, by replacing $a$ by $y\mapsto
a(y)+\nu\cos(2\pi\langle k,y\rangle)\eta_{n}\otimes\eta_{n}$ for some
$k\in\mathbb{Z}^{d}\setminus\langle e\rangle$ and $\nu\in\mathbb{R}$
sufficiently small.

Step 2: Restrict attention to $a_{e}^{\perp}$

First, observe that $\tilde{\mu}^{\eta_{n}}_{e}=\tilde{\mu}^{\eta_{m}}_{e}$ if
and only if there is a $C>0$ such that $\langle
a_{e}^{\perp}\eta_{n},\eta_{n}\rangle=C\langle
a_{e}^{\perp}\eta_{m},\eta_{m}\rangle$ in $\mathbb{T}^{d}$. This is a direct
consequence of the formula (18) proved in the previous section. We claim that
if $\langle a_{e}^{\perp}\eta_{n},\eta_{n}\rangle=C\langle
a_{e}^{\perp}\eta_{m},\eta_{m}\rangle$ for some $C>0$, then this symmetry is
broken by some arbitrarily small perturbation of $a$.

To see this, we will differentiate the function $a\mapsto a_{e}^{\perp}$.
Given $a_{*}\in C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d})$ and
$h\in\mathbb{R}$ small enough, define $a_{h}=a+ha_{*}$ and let
$(a_{h})_{e}^{\perp}$ be the associated averaged tensor.

Employing arguments similar to those in the proof of Proposition 7, we see
that, for each $r\in[0,r_{e})$, $h\in\mathbb{R}$, and $j\in\\{n,m\\}$, if
$\tilde{U}_{j,re}^{h}$ is the solution of the equation

$\left\\{\begin{array}[]{r
l}-\text{tr}\left(a_{h}(re+x^{\prime})D^{2}\tilde{U}_{j,re}^{h}\right)=\langle
a_{h}(re+x^{\prime})\eta_{j},\eta_{j}\rangle-\langle(a_{h})_{e}^{\perp}(re)\eta_{j},\eta_{j}\rangle&\text{in}\,\,\mathbb{T}^{d-1}_{e}(0)\\\
\tilde{U}_{j,re}^{h}(0)=0\end{array}\right.$

then there are functions $\\{\tilde{U}_{n,re},\tilde{U}_{m,re}\\}\in
C(\mathbb{T}^{d-1}_{e}(0))$ and a function $a_{*}^{\perp}\in
C(\mathbb{T}^{d})$ varying only in the $e$ direction such that, for
$j\in\\{n,m\\}$ and $r\in[0,r_{e})$,

$\tilde{U}_{j,re}=\lim_{h\to
0}\frac{\tilde{U}^{h}_{j,re}-\tilde{U}^{0}_{j,re}}{h}\quad\text{uniformly
in}\,\,\mathbb{T}^{d-1}_{e}(0),\quad a_{*}^{\perp}=\lim_{h\to
0}\frac{(a_{h})_{e}^{\perp}-a_{e}^{\perp}}{h}\quad\text{uniformly
in}\,\,\mathbb{T}^{d}.$

Furthermore, $\tilde{U}_{j,re}$ is a solution of the equation

$-\text{tr}\left(a(re+x^{\prime})D^{2}_{e}\tilde{U}_{j,re}\right)=\text{tr}\left(a_{*}(re+x^{\prime})D^{2}_{e}u_{j}(re+x^{\prime})\right)-\langle
a_{*}^{\perp}(re)\eta_{j},\eta_{j}\rangle\quad\text{in}\,\,\mathbb{T}^{d-1}_{e}(0).$

Notice that
$a_{*}^{\perp}(y)=\int_{\mathbb{T}^{d}}\text{tr}\left(a_{*}(y^{\prime})D^{2}_{e}u_{j}(y^{\prime})\right)\mu_{e}^{\langle
y,e\rangle}(dy^{\prime})$.

By the previous step, there is no loss in generality assuming that
$D^{2}_{e}u_{n}\neq D^{2}_{e}u_{m}$. In fact, since these functions are
$C^{\alpha}$, we can fix $y_{1},y_{2}\in\mathbb{T}^{d}$ so that $\langle
y_{1},e\rangle\neq\langle y_{2},e\rangle$ and $D^{2}_{e}u_{n}(y_{i})\neq
D^{2}_{e}u_{m}(y_{i})$ for $i\in\\{1,2\\}$. Since $\mu_{e}^{s}$ is supported
on $\mathbb{T}^{d-1}_{e}(s)$ for each $s\in[0,r_{e})$, we can we fix $a_{*}\in
C^{\infty}(\mathbb{T}^{d};\mathcal{S}_{d})$ so that

$\displaystyle\int_{\mathbb{T}^{d}}\text{tr}\left(a_{*}(y^{\prime})D^{2}_{e}u_{n}(y^{\prime})\right)\mu_{e}^{\langle
y_{1},e\rangle}(dy^{\prime})>0>\int_{\mathbb{T}^{d}}\text{tr}\left(a_{*}(y^{\prime})D^{2}_{e}u_{m}(y^{\prime})\right)\,\mu_{e}^{\langle
y_{1},e\rangle}(dy^{\prime}),$
$\displaystyle\int_{\mathbb{T}^{d}}\text{tr}\left(a_{*}(y^{\prime})D^{2}_{e}u_{n}(y^{\prime})\right)\mu_{e}^{\langle
y_{2},e\rangle}(dy^{\prime})<0<\int_{\mathbb{T}^{d}}\text{tr}\left(a_{*}(y^{\prime})D^{2}_{e}u_{m}(y^{\prime})\right)\,\mu_{e}^{\langle
y_{2},e\rangle}(dy^{\prime}).$

From this, we find that, for all $h>0$ sufficiently small,

$\frac{\langle(a_{h})^{\perp}_{e}(y_{1})\eta_{m},\eta_{m}\rangle}{\langle(a_{h})^{\perp}_{e}(y_{1})\eta_{n},\eta_{n}\rangle}<C<\frac{\langle(a_{h})^{\perp}_{e}(y_{2})\eta_{m},\eta_{m}\rangle}{\langle(a_{h})^{\perp}_{e}(y_{2})\eta_{n},\eta_{n}\rangle}.$

Therefore, $\langle(a_{h})_{e}^{\perp}\eta_{n},\eta_{n}\rangle$ is not a
constant multiple of $\langle(a_{h})_{e}^{\perp}\eta_{m},\eta_{m}\rangle$ even
while $\|a-a_{h}\|_{C^{2,\alpha}(\mathbb{T}^{d})}\leq Ch$.

We conclude from the preceding that $\mathcal{U}_{e}(n,m)$ is dense in
$C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))$. It remains to
prove the same thing for $\mathcal{V}_{e}(n,m)$.

Step 3: Ensure $\tilde{a}^{\eta_{n}}_{e}\neq\tilde{a}^{\eta_{m}}_{e}$

We want to show that $\mathcal{V}_{e}(n,m)$ is dense. Given that
$\tilde{a}^{\eta}_{e}=\int_{\mathbb{T}^{d}}a(y)\,\tilde{\mu}^{\eta}_{e}(dy)$,
this is intuitively clear given what we just proved.

The previous arguments show that we can assume that
$a\in\mathcal{U}_{e}(n,m)\setminus\mathcal{V}_{e}(n,m)$ to start with. We will
show that there is an $m\in C^{2,\alpha}(\mathbb{T}^{d})$ varying only in the
$e$ direction such that the function $a_{h}=(1+hm)^{-1}a$ is in
$\mathcal{V}_{e}(n,m)$ for all $h\in\mathbb{R}$ sufficiently small.

To start with, notice that, by arguing as in Section 4.4, we see that, for
each $\eta\in S^{d-1}\cap\langle e\rangle^{\perp}$, the limiting coefficient
$\tilde{a}_{h,e}^{\eta}$ associated with $a_{h}$ is given by

(63)
$\tilde{a}_{h,e}^{\eta}=\frac{\tilde{a}_{e}^{\eta}}{1+h\int_{\mathbb{T}^{d}}m(y)\tilde{\mu}^{\eta}_{e}(dy)}.$

Since $m$ only varies in the $e$ direction, the integral in the denominator
becomes

$\int_{\mathbb{T}^{d}}m(y)\tilde{\mu}^{\eta}_{e}(dy)=\frac{\int_{0}^{r_{e}}m(se)\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds}{\int_{0}^{r_{e}}\langle
a_{e}^{\perp}(se)\eta,\eta\rangle^{-1}\,ds}.$

Given that $a\in\mathcal{U}_{e}(n,m)$, we know that $\langle
a_{e}^{\perp}(se)\eta_{n},\eta_{n}\rangle$ does not equal a multiple of
$\langle a_{e}^{\perp}(se)\eta_{m},\eta_{m}\rangle$, and, therefore, we can
choose $m$ so that $\int_{\mathbb{T}^{d}}m(\langle y,e\rangle
e)\tilde{\mu}^{\eta_{n}}_{e}(dy)\neq\int_{\mathbb{T}^{d}}m(\langle y,e\rangle
e)\tilde{\mu}^{\eta_{m}}_{e}(dy)$. Therefore, since
$\tilde{a}^{\eta_{n}}_{e}=\tilde{a}^{\eta_{m}}_{e}$, (63) implies
$\tilde{a}^{\eta_{n}}_{h,e}\neq\tilde{a}^{\eta_{m}}_{h,e}$ for all $h$ small
enough. This proves $a$ is a limit point of $\mathcal{V}_{e}(n,m)$.

Conclusion

We showed that $\mathcal{U}_{e}(n,m)$ and $\mathcal{V}_{e}(n,m)$ are both open
and dense in $C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))$
in the $C^{2,\alpha}$ norm topology. Define $\mathscr{C}_{d}$ by

$\mathscr{C}_{d}=\bigcap_{n\in\mathbb{N}}\bigcap_{m\in\mathbb{N}\setminus\\{n\\}}\mathcal{U}_{e}(n,m)\cap\mathcal{V}_{e}(n,m).$

This set is residual, being a countable intersection of open, dense sets.
Further, since $C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d}(\lambda,\Lambda))$
is an open subset of the Banach space
$C^{2,\alpha}(\mathbb{T}^{d};\mathcal{S}_{d})$, $\mathscr{C}_{d}$ is itself
dense. ∎

##  7\. Effective Mobility as Linear Response

In this section, we prove Theorems 5 and 6 and their corollaries. As already
indicated in the introduction, the proofs are relatively routine when
$e\notin\mathbb{R}\mathbb{Z}^{d}$. We will start by motivating the
modifications that are necessary in the case that
$e\in\mathbb{R}\mathbb{Z}^{d}$.

###  7.1. Strategy of proof

As already mentioned in Section 2, the analysis of the forced problem (20)
requires an additional correction. Fix $e\in\mathbb{R}\mathbb{Z}^{d}$,
$\alpha\in\mathbb{R}\setminus\\{0\\}$, and let
$(u^{\epsilon}_{e})_{\epsilon>0}$ be the solutions of (20). Proceeding as in
the proof of Theorems 1 and 2, we seek to identify the limit of
$(u^{\epsilon}_{e})_{\epsilon>0}$ by showing that the functions
$\bar{u}^{*}_{e}$ and $\bar{u}_{*,e}$ are sub- and supersolutions of an
effective equation.

Let us see what can be deduced naïvely about $\bar{u}^{*}_{e}$ using the
perturbed test function method. As in Section 4.5, we let $\tilde{V}_{e}$ be
the solution of the following cell problem:

(64)
$m(y,e)-m_{e}^{\perp}(y)-\text{tr}\left(A(y,e)D^{2}\tilde{V}_{e}\right)=0\quad\text{in}\,\,\mathbb{T}^{d}.$

Suppose that $(x_{0},t_{0})\in\mathbb{R}^{d}\times(0,\infty)$, $\varphi$ is
smooth, and $\bar{u}^{*}_{e}-\varphi$ has a strict local maximum at
$(x_{0},t_{0})$. Since we expect that $\bar{u}^{*}_{e}(x,t)=\langle
x,e\rangle+\alpha\overline{m}_{pl}(e)t$ for some $\overline{m}_{pl}(e)>0$, we
may as well assume that $\widehat{D\varphi}(x_{0},t_{0})=e$.

With $\tilde{V}_{e}$ a solution of (64), define the perturbed test function
$\varphi^{\epsilon}$ by

$\varphi^{\epsilon}(x,t)=\varphi(x,t)+\epsilon^{2}\varphi_{t}(x_{0},t_{0})\tilde{V}_{e}(\epsilon^{-1}x).$

Letting $(x_{\epsilon},t_{\epsilon})$ be a local maximum of
$u^{\epsilon}_{e}-\varphi^{\epsilon}$ close enough to $(x_{0},t_{0})$, we know
that $(x_{\epsilon},t_{\epsilon})\to(x_{0},t_{0})$ (along a subsequence) and,
thus, we can invoke the equation satisfied by $u^{\epsilon}_{e}$, which gives,
after some simplification,

$\limsup_{\epsilon\to 0^{+}}\left[m_{e}^{\perp}(\epsilon^{-1}\langle
x_{\epsilon},e\rangle)\varphi_{t}^{\epsilon}(x_{\epsilon},t_{\epsilon})-\alpha\|D\varphi^{\epsilon}(x_{\epsilon},t_{\epsilon})\|\right]\leq
0.$

This shows that to homogenize $(u^{\epsilon}_{e})_{\epsilon>0}$, it is not
enough to treat the transversal fluctuations of the front using
$\tilde{V}_{e}$; the position of the front along the $e$ axis continues to
oscillate. Nonetheless, we have lost $(d-1)$-degrees of freedom by averaging,
and the equation obtained so far suggests that we should be able to proceed by
exploiting the asymptotic behavior of the averaged, one-dimension equation:

(65) $\left\\{\begin{array}[]{r
l}m_{e}^{\perp}(se)\mathcal{U}_{t}-\alpha|\mathcal{U}_{s}|=0&\text{in}\,\,\mathbb{R}\times(0,\infty),\\\
\mathcal{U}(s,0)=s&\text{if}\,\,s\in\mathbb{R}.\end{array}\right.$

This is precisely the approach taken in what follows.

###  7.2. Intermediate results

Let us state precisely some of the results used in the proof of Theorem 5.

First, we construct traveling wave sub- and supersolutions of (20) in rational
directions:

######  Proposition 17.

Under the same assumptions as in Theorems 2 and 5, if
$e\in\mathbb{R}\mathbb{Z}^{d}$ and $\alpha\in\mathbb{R}\setminus\\{0\\}$,
then:

  * (i)

The cell problem (64) has a solution $\tilde{V}_{e}\in
C^{2,\alpha}(\mathbb{T}^{d})$ and $m^{\perp}_{e}\in
C^{2,\alpha}(\mathbb{T}^{d})$.

  * (ii)

There is a $C^{2}$, $r_{e}$-periodic function $P_{e}:\mathbb{R}\to\mathbb{R}$
and a constant $\overline{m}_{pl}(e)$ such that

$\overline{m}_{pl}(e)^{-1}m_{e}^{\perp}(se)-|1+P_{e,s}|=0\quad\text{in}\,\,\mathbb{R}.$

In particular, for each $\alpha\in\mathbb{R}\setminus\\{0\\}$, the function
$(s,t)\mapsto s+P_{e}(s)+\alpha\overline{m}_{pl}(e)^{-1}t$ is a pulsating wave
solution of (65). Moreover, $\overline{m}_{pl}(e)$ is given explicitly by
(22).

  * (iii)

For each $\alpha\in\mathbb{R}\setminus\\{0\\}$, there is an
$\epsilon_{0}^{+}>0$ and a family
$(\alpha^{+}_{\epsilon})_{\epsilon\in(0,\epsilon_{0}^{+})}$ such that the
functions $(u^{+,\epsilon})_{\epsilon\in(0,\epsilon_{0}^{+})}$ defined by

$u^{+,\epsilon}(x,t)=\langle x,e\rangle+\epsilon P_{e}(\epsilon^{-1}\langle
x,e\rangle)+\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}\left(\epsilon^{2}\tilde{V}_{e}(\epsilon^{-1}x)+t\right)$

are super-solutions of (20). Furthermore, there is a constant $C_{+}>0$
depending only on $\tilde{V}_{e}$ and $P_{e}$ such that, for each
$(x,t)\in\mathbb{R}^{d}\times\mathbb{R}$,

$\displaystyle|\alpha^{+}_{\epsilon}-\alpha|$ $\displaystyle\leq
C_{+}\epsilon$ $\displaystyle|u^{+,\epsilon}(x,t)-\langle
x,e\rangle-\alpha\overline{m}(e)^{-1}t|$ $\displaystyle\leq
C_{+}\epsilon(1+|t|)$

  * (iv)

For each $\alpha\in\mathbb{R}\setminus\\{0\\}$, there is an
$\epsilon_{0}^{-}>0$ and a family
$(\alpha^{-}_{\epsilon})_{\epsilon\in(0,\epsilon_{0}^{-})}$ such that the
functions $(u^{-,\epsilon})_{\epsilon\in(0,\epsilon_{0}^{-})}$ defined by

$u^{-,\epsilon}(x,t)=\langle x,e\rangle+\epsilon P_{e}(\epsilon^{-1}\langle
x,e\rangle)+\alpha_{\epsilon}^{-}\overline{m}_{pl}(e)^{-1}\left(\epsilon^{2}\tilde{V}_{e}(\epsilon^{-1}x)+t\right)$

are sub-solutions of (20) in $\mathbb{R}^{d}\times\mathbb{R}$. Furthermore,
there is a constant $C_{-}>0$ depending only on $\tilde{V}_{e}$ and $P_{e}$
such that, for each $(x,t)\in\mathbb{R}^{d}\times\mathbb{R}$,

$\displaystyle|\alpha^{-}_{\epsilon}-\alpha|$ $\displaystyle\leq
C_{-}\epsilon$ $\displaystyle|u^{-,\epsilon}(x,t)-\langle
x,e\rangle-\alpha\overline{m}(e)^{-1}t|$ $\displaystyle\leq
C_{-}\epsilon(1+|t|)$

A similar construction works when $e\notin\mathbb{R}\mathbb{Z}^{d}$ and (64)
has a $C^{2}$ solution. Here is the result in that case:

######  Proposition 18.

If there is a solution $V_{e}\in C^{2}(\mathbb{T}^{d})$ of (64) and $e\in
S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$, then there is an $\epsilon_{0}>0$
and families $(\alpha_{\epsilon}^{+})_{\epsilon\in(0,\epsilon_{0})}$ and
$(\alpha_{\epsilon}^{-})_{\epsilon\in(0,\epsilon_{0})}$ such that the
functions $(u^{+,\epsilon})_{\epsilon\in(0,\epsilon_{0})}$ and
$(u^{-,\epsilon})_{\epsilon\in(0,\epsilon_{0})}$ defined by

$u^{\pm,\epsilon}(x,t)=\langle
x,e\rangle+\alpha_{\epsilon}^{\pm}\overline{m}(e)^{-1}\left(\epsilon^{2}V_{e}(\epsilon^{-1}x)+t\right)$

are, respectively, super- and sub-solutions of (20) in
$\mathbb{R}^{d}\times\mathbb{R}$. Furthermore, there is a $C>0$ such that, for
each $(x,t)\in\mathbb{R}^{d}\times\mathbb{R}$,

$\displaystyle|\alpha^{+}_{\epsilon}-\alpha|+|\alpha^{-}_{\epsilon}-\alpha|$
$\displaystyle\leq C\epsilon$ $\displaystyle|u^{+,\epsilon}(x,t)-\langle
x,e\rangle-\alpha\overline{m}(e)^{-1}t|+|u^{-,\epsilon}(x,t)-\langle
x,e\rangle-\alpha\overline{m}(e)^{-1}t|$ $\displaystyle\leq C\epsilon(1+|t|)$

Since the proof of Proposition 18 is effectively a simpler version of that of
Proposition 17, we omit it. In general, it is far from clear whether or not
(64) has a solution when $e\notin\mathbb{R}\mathbb{Z}^{d}$ due to the loss of
compactness; cases where this is possible are discussed in Appendix A.

When applicable, Propositions 17 and 18 allow us to quantify the convergence
in Theorem 5. More precisely, we have

######  Proposition 19.

If $e\in S^{d-1}\cap\mathbb{R}\mathbb{Z}^{d}$ or there is a $V_{e}\in
C^{2}(\mathbb{T}^{d})$ solving (23), then there is an $\epsilon_{0}\in(0,1)$
and a $C_{e}>0$ such that if $\epsilon\in(0,\epsilon_{0})$ and
$u_{e}^{\epsilon}$ solves (20) with $u_{e}^{\epsilon}(x,0)=\langle
x,e\rangle$, then

$|u_{e}^{\epsilon}(x,t)-\langle
x,e\rangle-\alpha\overline{m}_{pl}(e)^{-1}t|\leq
C_{e}\epsilon(1+t)\quad\text{if}\,\,(x,t)\in\mathbb{R}^{d}\times(0,\infty)$

Since the proof of Proposition 19 follows directly from Propositions 17 and 18
and the comparison principle, we omit it.

It only remains to consider the case when $e\notin\mathbb{R}\mathbb{Z}^{d}$
yet the hypotheses of Proposition 18 fail. Using approximate correctors, it is
still possible to construct traveling wave sub- and supersolutions and use
these to give a qualitative proof of Theorem 5. This is briefly treated in
Section 7.4 below.

###  7.3. Proof of Proposition 17

Most of the work in proving the proposition lies in (iii) and (iv) since (i)
is Proposition 7 and (ii) is an exercise in ODE theory. More precisely, (ii)
is covered by the following lemma, the proof of which is left to the
interested reader:

######  Lemma 1.

Fix $r>0$. If $\tilde{m}:r\mathbb{T}\to(0,\infty)$ is $C^{1}$, then the
solution $X^{s,\tilde{m}}$ of the one-dimensional ODE

$\left\\{\begin{array}[]{r
l}\tilde{m}(X^{s,\tilde{m}}_{t})\dot{X}^{s,\tilde{m}}_{t}=1,\\\
X^{s,\tilde{m}}_{0}=s,\end{array}\right.$

satisfies

$\lim_{t\to\infty}\frac{X^{s,\tilde{m}}_{t}-s}{t}=\frac{1}{\fint_{0}^{r}\tilde{m}(u)\,du}.$

Furthermore, the function $\tilde{P}:r\mathbb{T}\to\mathbb{R}$ given by

$\tilde{P}(s)=\left(\fint_{0}^{r_{e}}\tilde{m}(u)\,du\right)^{-1}\int_{0}^{s}\tilde{m}(s)\,ds-1$

solves the pulsating wave equation

$\frac{\tilde{m}(s)}{\fint_{0}^{r}\tilde{m}(u)\,du}-|1+P^{\prime}(s)|=0\quad\text{in}\,\,r\mathbb{T}.$

Now we proceed with the

###### Proof of Proposition 17.

(i) follows from Proposition 7 above, and (ii) is an application of Lemma 1.
We will only prove (iii) since (iv) follows similarly.

Let $\epsilon>0$ and $\alpha_{\epsilon}^{+}\in\mathbb{R}$ be free variables
for the moment and define
$u^{+,\epsilon}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ by

(66) $u^{+,\epsilon}(x,t)=\langle x,e\rangle+\epsilon
P_{e}(\epsilon^{-1}\langle
x,e\rangle)+\alpha_{\epsilon}\overline{m}_{pl}(e)^{-1}\left(\epsilon^{2}\tilde{V}_{e}(\epsilon^{-1}x)+t\right).$

We will show that if $\epsilon>0$ is small enough and
$\alpha_{\epsilon}^{+}=\alpha+C_{0}\epsilon$ for some large enough constant
$C_{0}>0$, then $u^{+,\epsilon}$ is a super-solution of (20).

Let us study the equation for $u^{+,\epsilon}$ term by term. First, the term
with the time derivative. To declutter the notation, we define
$p_{\epsilon}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ by

$p_{\epsilon}(x,t)=Du^{+,\epsilon}(x,t)=(1+P^{\prime}_{e}(\epsilon^{-1}\langle
x,e\rangle))e+\epsilon\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}D\tilde{V}_{e}(\epsilon^{-1}x).$

Observe that since $1+P^{\prime}_{e}>0$, it follows that
$\hat{p_{\epsilon}}=e+O(\epsilon)$. Thus,

$\displaystyle
m(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})u^{+,\epsilon}_{t}=\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}m(\epsilon^{-1}x,e)+O(\epsilon).$

Next, the curvature term. First, notice that if we write
$\mathcal{N}(p)=\hat{p}\otimes\hat{p}$, then, by Taylor expansion,

$\displaystyle\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})D^{2}u^{+,\epsilon}\right)$
$\displaystyle=\text{tr}\left(a(\epsilon^{-1}x,p_{\epsilon})(\text{Id}-\hat{p}_{\epsilon}\otimes\hat{p}_{\epsilon})D^{2}u^{+,\epsilon}(\text{Id}-\hat{p}_{\epsilon}\otimes\hat{p}_{\epsilon})\right)$
$\displaystyle=\text{tr}\left(a(\epsilon^{-1}x,e)(\text{Id}-e\otimes
e)D^{2}u^{+,\epsilon}(\text{Id}-e\otimes e)\right)$
$\displaystyle\quad+\text{tr}\left(D_{p}a(\epsilon^{-1}x,e)[p_{\epsilon}-e](\text{Id}-e\otimes
e)D^{2}u^{+,\epsilon}(\text{Id}-e\otimes e)\right)$
$\displaystyle\quad+\text{tr}\left(a(\epsilon^{-1}x,e)D_{p}\mathcal{N}(e)[p_{\epsilon}-e]D^{2}u^{+,\epsilon}(\text{Id}-e\otimes
e)\right)$
$\displaystyle\quad+\text{tr}\left(a(\epsilon^{-1}x,e)(\text{Id}-e\otimes
e)D^{2}u^{+,\epsilon}D_{p}\mathcal{N}(e)[p_{\epsilon}-e]\right)$
$\displaystyle\quad+\|D^{2}u^{+,\epsilon}\|_{L^{\infty}(\mathbb{T}^{d})}O(\|\hat{p}_{\epsilon}-e\|^{2})$

Next, observe that
$D^{2}u^{+,\epsilon}=\epsilon^{-1}P_{e}^{\prime\prime}e\otimes
e+\alpha_{\epsilon}\overline{m}_{pl}(e)^{-1}D^{2}\tilde{V}_{e}$ and, thus, in
each of the first four terms in the previous expressions, we can replace
$D^{2}u^{+,\epsilon}$ by
$\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}D^{2}\tilde{V}_{e}$. This leads
to

$\displaystyle\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})D^{2}u^{+,\epsilon}\right)$
$\displaystyle=\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}\text{tr}\left(A(\epsilon^{-1}x,e)D^{2}\tilde{V}_{e}\right)+\|D^{2}\tilde{V}_{e}\|_{L^{\infty}(\mathbb{T}^{d})}O(\|\hat{p}_{\epsilon}-e\|)$
$\displaystyle\quad+\|D^{2}u^{+,\epsilon}\|_{L^{\infty}(\mathbb{T}^{d})}O(\|\hat{p}_{\epsilon}-e\|^{2})$

Since $\|D^{2}u^{+,\epsilon}\|\leq C\epsilon^{-1}$ and
$\|\hat{p}_{\epsilon}-e\|=O(\epsilon)$, we conclude

$\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})D^{2}u^{+,\epsilon}\right)=\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}\text{tr}\left(A(\epsilon^{-1}x,e)D^{2}V_{e}\right)+O(\epsilon).$

Finally, we treat the first-order term. Since $\|e\|=1$, we have

$\displaystyle\alpha\|Du^{+,\epsilon}\|$
$\displaystyle=\alpha|1+P_{e}^{\prime}|+O(\epsilon).$

Putting it all together and invoking the equations satisfied by
$\tilde{V}_{\epsilon}$ and $P_{e}$, we find, for some $\overline{C}>0$,

$\displaystyle
m(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})u^{+,\epsilon}_{t}-\text{tr}\left(A(\epsilon^{-1}x,\widehat{Du^{+,\epsilon}})D^{2}u^{+,\epsilon}\right)-\alpha\|Du^{+,\epsilon}\|\geq$
$\displaystyle\qquad\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}\left(m(\epsilon^{-1}x,e)-\text{tr}\left(A(\epsilon^{-1}x,e)D^{2}\tilde{V}_{e}\right)\right)-\alpha|1+P_{e}^{\prime}|-(\overline{C}+o(1))\epsilon=$
$\displaystyle\qquad\alpha_{\epsilon}^{+}\overline{m}_{pl}(e)^{-1}m_{e}^{\perp}(\epsilon^{-1}\langle
x,e\rangle)-\alpha\overline{m}_{pl}(e)^{-1}m_{e}^{\perp}(\epsilon^{-1}\langle
x,e\rangle)-(\overline{C}+o(1))\epsilon$
$\displaystyle\qquad=(C_{0}-C+o(1))\epsilon\overline{m}_{pl}(e)^{-1}m_{e}^{\perp}(\epsilon^{-1}\langle
x,e\rangle).$

Setting $C_{0}=C+1$, we deduce that there is an $\epsilon_{0}>0$ such that
$u^{+,\epsilon}$ is a super-solution of (20) in
$\mathbb{R}^{d}\times\mathbb{R}$.∎

###  7.4. Proofs of Theorems 5 and 6

Finally, we complete the proof of Theorem 5 and address Theorem 6.

In view of Propositions 17 and 18, Theorem 5 is proved as soon as we analyze
the case that $e\notin\mathbb{R}\mathbb{Z}^{d}$ and yet (64) has no smooth
solution. Not surprisingly, we argue using approximate correctors; the
argument is included only for the sake of completeness.

Theorem 6 follows directly from Proposition 17 when
$e\in\mathbb{R}\mathbb{Z}^{d}$ and the proof of Theorem 5, otherwise.

###### Proof of Theorem 5.

Assume $e\in S^{d-1}\setminus\mathbb{R}\mathbb{Z}^{d}$; otherwise, Proposition
17 applies. Given $\delta>0$, let $V^{\delta}$ be the solution of (39) with
$f(y)=-m(y,e)$. By Schauder estimates, there is a constant $C>0$ such that

(67)
$\delta\|V^{\delta}\|_{L^{\infty}(\mathbb{T}^{d})}+\delta^{2}\|DV^{\delta}\|_{L^{\infty}(\mathbb{T}^{d})}+\delta^{3}\|D^{2}V^{\delta}\|_{L^{\infty}(\mathbb{T}^{d})}\leq
C.$

To start with, we claim that

(68) $\limsup\nolimits^{*}u_{e}^{\epsilon}(x,t)\leq\langle
x,e\rangle+\alpha\overline{m}(e)^{-1}t.$

To see this, fix $\beta\in(\alpha,\infty)$, let $\epsilon\in(0,1)$, set
$\delta(\epsilon)=\epsilon^{\frac{1}{4}}$, and define
$v^{\epsilon}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ by

$v^{\epsilon}(x,t)=\langle
x,e\rangle+\beta\overline{m}(e)^{-1}\left(t+\epsilon^{2}V^{\delta(\epsilon)}_{e}(\epsilon^{-1}x)\right).$

We claim there is an $\epsilon_{0}>0$ depending only on $m$ and $\beta$ such
that $v^{\epsilon}$ is a super-solution of (20) if
$\epsilon\in(0,\epsilon_{0})$.

Indeed, invoking (67), we find

$\displaystyle m(\epsilon^{-1}x,\widehat{Dv^{\epsilon}})$ $\displaystyle
v^{\epsilon}_{t}-\text{tr}\left(A(\epsilon^{-1}x,\widehat{Dv^{\epsilon}})D^{2}v^{\epsilon}\right)-\alpha\|Dv^{\epsilon}\|$
$\displaystyle\quad=(\beta-\alpha)+\beta\overline{m}(e)^{-1}(-\delta
V^{\delta}_{e}(\epsilon^{-1}x)-\overline{m}(e))+O(\epsilon^{\frac{1}{2}})$
$\displaystyle\quad=\beta-\alpha+o(1).$