# Phase Transition of Parabolic Ginzburg–Landau Equation with Potentials of
High-Dimensional Wells

Yuning Liu  NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China, and
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan
Road North, Shanghai, 200062, China<EMAIL_ADDRESS>
###### Abstract.

In this work, we study the co-dimensional one interface limit and geometric
motions of parabolic Ginzburg–Landau systems with potentials of high-
dimensional wells. The main result generalizes the one by Lin et al. (Comm.
Pure Appl. Math., 65(6):833-888, 2012) to a dynamical case. In particular
combining modulated energy methods and weak convergence methods, we derive the
limiting harmonic heat flows in the inner and outer bulk regions segregated by
the sharp interface, and a non-standard boundary condition for them. These
results are valid provided that the initial datum of the system is well-
prepared under natural energy assumptions.

##  1\. Introduction

In the work of Keller–Rubinstein–Sternberg [32, 33], they created a general
gradient flow theory in the descriptions of fast reaction and slow diffusion,
and established its relations to the mean curvature flow (MCF) and the
harmonic heat flows into manifolds. These involve some formal statements
associated with the multiple component phase transitions with higher
dimensional wells. Such statements are also refereed to as the
Keller–Rubinstein–Sternberg problem. More precisely they investigated the
vectorial Allen–Cahn equation (also called Ginzburg–Landau equation)

$\partial_{t}\mathbf{u}_{\varepsilon}=\Delta\mathbf{u}_{\varepsilon}-\varepsilon^{-2}\partial
F(\mathbf{u}_{\varepsilon}),$ (1.1)

where $\mathbf{u}_{\varepsilon}(x,t):\Omega\times(0,T)\mapsto\mathbb{R}^{n}$
is a mapping depending on a small parameter $\varepsilon>0$ and
$\Omega\subset\mathbb{R}^{k}$ is a bounded domain with $C^{1}$ boundary. Here
$F(\mathbf{u})$ is a double equal-well potential with ground state being the
disjoint union of two hypersurfaces $\mathfrak{m}_{\pm}\subset\mathbb{R}^{n}$,
and $\partial F(\mathbf{u})$ is the differential of $F$ at $\mathbf{u}$. The
Keller–Rubinstein–Sternberg problem is concerned with the limiting behavior of
$\mathbf{u}_{\varepsilon}$ as $\varepsilon$ tends to zero. In the work of
Lin–Pan–Wang [24], they set up an analytic program to rigorously justify the
formally asymptotic analysis given in the aforementioned works of
Keller–Rubinstein–Sternberg, and give a complete resolution of the static
problem. To be more precise they considered the minimizers of the
Ginzburg–Landau functional

$\int_{\Omega}\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})\right)\,dx$
(1.2)

that satisfy well-prepared boundary conditions on $\partial\Omega$. They
established the co-dimensional one interface limit of (1.2), which essentially
generalized the $\Gamma$-convergence of Modica–Mortola [28] to the vectorial
cases (though they did not state their main theorems in such an abstract
manner). More importantly, they showed that the limit of
$\mathbf{u}_{\varepsilon}$ in the bulk region corresponds to the harmonic maps
into $\mathfrak{m}_{\pm}$, and they derived the so called minimal pair
boundary condition which services as a constraint on the limiting harmonic
maps when they are restricted on the interface. Such a non-standard boundary
condition is a new feature that arises due to minimization of surface tension
in vectorial cases. Note that such a condition holds trivially in the case of
scalar Allen-Cahn equation because the limiting mappings get stuck at the two
distinct points in $\mathbb{R}^{1}$.

The aim of this work is to solve the (dynamical) Keller–Rubinstein–Sternberg
problem when $\mathfrak{m}_{\pm}$ are two disjoint hypersurfaces in the target
space $\mathbb{R}^{n}$. More precisely, we prove the following statements:
Firstly, for well-prepared initial data, as $\varepsilon$ tends to $0$, the
solution gradients of (1.1) will undergo phase transitions across a moving
interface $\Sigma_{t}$ that propagates as a (two-phase) mean curvature flow.
Secondly, in the two bulk regions $\Omega_{t}^{\pm}$ segregated by the
interface $\Sigma_{t}$, the solutions will converge to harmonic heat flows
mapping into $\mathfrak{m}_{\pm}$ respectively. Finally, the one-sided traces
of the limiting harmonic heat flows on $\Sigma_{t}$ must satisfy the minimal
pair boundary condition.

Our first result is a vectorial analogy of the co-dimensional one scaling
limit of scalar parabolic Allen–Cahn equation to the (two-phase) mean
curvature flow, i.e. the special case of (1.1) when $\mathfrak{m}_{\pm}$ are
two distinct points $a^{\pm}\in\mathbb{R}^{1}$. There have been major
progresses in the scalar case over the last 30 years, made under different
frameworks. Here we mention two classes of results and will discuss some other
classes in the sequel. One is the convergence to a Brakke’s flow by Ilmanen
[19] using a version of Huisken’s monotonicity formula together with tools
from geometric measure theory. See also [6, 18, 31, 30, 34] and the references
therein for further renovations. Despite of its energetic nature, a major
difficulty of such an approach is the control of the so called discrepancy
measure, and in every existing literature in this direction the solution
relies crucially on a version of Modica’s maximum principle [27]. There have
been attempts to generalize such a method to the vectorial cases. However, it
is not clear whether Modica’s maximum principle holds for elliptic system, cf.
[29]. Another approach, which relies more on the parabolic comparison
principle, is the convergence to the viscosity solution of mean curvature
flow. These are weak solutions to the mean curvature flow built independently
by Chen–Giga-Goto [7] and Evans–Spruck [12]. Concerning the global in time
convergence of scalar Allen–Cahn equation to such solutions, we refer the
readers to the work of Evans–Soner–Souganidis [11], the work of Soner [35] and
the references therein. These two approaches both give global in time (weak)
convergences to weakly defined solutions of MCFs up to their life spans.
However, as their technics involve parabolic maximum principle and the
comparison principle in one way or another, it is not clear how to use them to
attack the vectorial cases in general. It is worth mentioning that for
radially symmetric initial data and when $\mathfrak{m}_{\pm}$ are two
concentric circles, Bronsard–Stoth [4] obtain global in time convergences to
MCF of planar circles.

To the best of our knowledge, there are mainly two approaches to rigorously
justify the convergences of the vectorial Allen–Cahn equations, both assuming
the limiting interface propagation problem has a (local in time) classical
solution. Compared with the aforementioned methods which lead to global in
time (weak) convergences, they have quite different natures. One of these
methods is the asymptotical expansion technics developed by De
Mottoni–Schatzman [9] and Alikakos–Bates–Chen [1], which has been used
recently in [13, 14] for matrix-valued cases of (1.1). In particular,
Fei–Lin–Wang–Zhang [13] studied the case when $\mathfrak{m}_{\pm}=O^{\pm}(n)$,
the $n$-dimensional orthogonal group, and they derived the minimal pair
boundary condition in a constructive way. By inner-outer expansions together
with a gluing procedure, such an approach reduces the convergence problem to a
linear stability problem given that the limiting system (not merely the
limiting interface motion) is strongly well-posed. We refer Lin–Wang [25] for
a general theory for the strong well-posedness of the limiting system. The
major challenge of this approach is the analysis of the spectrum of the
linearized operator at the so called ‘optimal profile’. Indeed, the spectrum
estimate in [13] makes delicate use of decompositions involving the geometry
of $O(n)$ and is much trickier than the scalar case obtained previously in [9,
5].

Another approach, which also assumes a regular solution of the limiting
interface motion but not the limiting harmonic heat flows, is the relative
entropy method developed by Fischer–Laux–Simon [16], motivated by
Jerrard–Smets [15] and Fischer–Hensel [20]. A generalization to matrix–valued
case has been done by Laux-Liu [21] to study the isotropic–nematic transition
in Landau–De Gennes model of liquid crystals, which essentially corresponds to
the case when $\mathfrak{m}_{+}=\mathbb{RP}^{2}$ and $\mathfrak{m}_{-}=0$.
More recently, in [26] the author used these methods, together with those
developed recently by Lin–Wang [23], to attack the convergence problem of an
anisotropic 2D Ginzburg–Landau model. In particular, he derived some delicate
convergence results of the level sets of the solutions, which are crucial to
obtain anchoring boundary conditions of the limiting solutions on the moving
interface.

Now we introduce a minimum amount of terminologies necessary for stating the
main result of this work. Let

$\mathfrak{m}_{\pm}\text{ be two disjoint smooth, closed, connected
hypersurfaces in }\mathbb{R}^{n}.$ (1.3)

For technical purposes we assume $0\in\mathfrak{m}_{-}$. We assume that
$F:\mathbb{R}^{n}\to[0,\infty)$ is a smooth function with $\mathfrak{m}_{\pm}$
being its double equal-wells:

$\operatorname{Arg~{}min}F=\mathfrak{m}:=\mathfrak{m}_{+}\sqcup\mathfrak{m}_{-}.$
(1.4)

We assume that $F(\mathbf{u})$ only depends on the distance from $\mathbf{u}$
to $\mathfrak{m}$. That is,

$F(\mathbf{u})=f(\mathrm{d}_{\mathfrak{m}}^{2}(\mathbf{u})),$ (1.5)

where $\mathrm{d}_{\mathfrak{m}}(\mathbf{u})$ is the signed-distance (see
(2.17) below for the full definition), and $f$ satisfies

$\displaystyle\begin{cases}f(s)\in C^{2}(\mathbb{R}^{+},\mathbb{R}^{+}),&\\\
c_{1}s\leq f(s)\leq c_{2}s&\text{ if }0\leq s\leq\delta_{0}^{2},\\\
f(s)=c_{3}&\text{ if }s\geq\delta_{0}^{2},\\\ \lim_{s\downarrow
0}\tfrac{f(s)}{s}=c_{4},\end{cases}$ (1.6)

for fixed positive constants
$c_{1},c_{2},c_{3},c_{4},\delta_{0}\in\mathbb{R}^{+}$. Here $\delta_{0}$ is a
small number so that the nearest-point projection $P_{\mathfrak{m}}$ from
$B_{2\delta_{0}}(\mathfrak{m})$, the $2\delta_{0}$-tubular neighborhood of
$\mathfrak{m}$, to $\mathfrak{m}$ is smooth.

We consider the following initial boundary value problems on a bounded domain
$\Omega\subset\mathbb{R}^{d}$ with $C^{1}$ boundary:

$\displaystyle\partial_{t}\mathbf{u}_{\varepsilon}$
$\displaystyle=\Delta\mathbf{u}_{\varepsilon}-\varepsilon^{-2}\partial
F(\mathbf{u}_{\varepsilon})$ $\displaystyle~{}\text{in}~{}\Omega\times(0,T),$
(1.7a) $\displaystyle\mathbf{u}_{\varepsilon}$
$\displaystyle=\mathbf{u}_{\varepsilon}^{in},$
$\displaystyle~{}\text{in}~{}\Omega\times\\{0\\},$ (1.7b)
$\displaystyle\mathbf{u}_{\varepsilon}$ $\displaystyle=\mathbf{g},$
$\displaystyle~{}\text{on}~{}\partial\Omega\times(0,T).$ (1.7c)

Here $\partial F(\mathbf{u})$ is the gradient of $F(\mathbf{u})$, and
$\mathbf{g}:\overline{\Omega}\mapsto\mathfrak{m}_{-}$ is a smoothing mapping.
Our main result is concerned with the asymptotical behaviors of solutions to
(1.7) for well-prepared initial datum. To give an analytic characterization of
such initial datum, we need to set up the geometry of the interface motion. To
this end, we assume

$\Sigma=\bigcup_{t\in[0,T]}\Sigma_{t}\times\\{t\\}~{}\text{is a smoothly
evolving closed hypersurface in}~{}\Omega,$ (1.8)

starting from a closed smooth surface $\Sigma_{0}\subset\Omega$. We denote by
$\Omega^{\pm}_{t}$ the domain enclosed by $\Sigma_{t}$, and by

$\displaystyle\mathrm{d}_{\Sigma}(x,t)\text{ the signed-distance from }x\text{
to the set }\Sigma_{t}\text{ taking positive values in }\Omega^{+}_{t}$ (1.9)

and negative values in
$\Omega^{-}_{t}=\Omega\backslash\overline{\Omega^{+}_{t}}$. In other words,

$\Omega^{\pm}_{t}:=\\{x\in\Omega\mid\mathrm{d}_{\Sigma}(x,t)\gtrless 0\\}.$
(1.10)

To avoid contact angle problems, we assume that $\Sigma$ stays at least
$\delta_{0}$ distant away from $\partial\Omega$.

Following [20, 15, 16], we define the modulated energy (also called the
relative entropy energy) by

$\displaystyle E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](t)$
$\displaystyle:=\int_{\Omega}\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}(\cdot,t)\right|^{2}+\frac{1}{\varepsilon}{F(\mathbf{u}_{\varepsilon}(\cdot,t))}-\bm{\xi}\cdot\nabla\psi_{\varepsilon}(\cdot,t)\right)\,dx.$
(1.11)

Here $\bm{\xi}$ is an appropriate extension of the unit normal vector field of
$\Sigma$ (see (2.28) below), and $\psi_{\varepsilon}$ is the scalar function

$\displaystyle\psi_{\varepsilon}(x,t):=\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}(x,t)$
(1.12)

with $\mathrm{d}_{F}$ being defined by (2.18) below. As we shall see later on,
the integrand of (1.11) is non-negative, and enjoys several coercivity
estimates including controls of the discrepancy and the deviations of the
normal vectors. We also need the surface tension coefficient

$c_{F}:=2\int_{0}^{\frac{1}{2}\mathrm{dist}_{\mathfrak{m}}}\sqrt{2f\left(\lambda^{2}\right)}d\lambda,$
(1.13)

where $\mathrm{dist}_{\mathfrak{m}}$ is the Euclidean distance between
$\mathfrak{m}_{+}$ and $\mathfrak{m}_{-}$, and another modulated energy
controlling the bulk errors:

$B[\mathbf{u}_{\varepsilon}|\Sigma](t):=\int_{\Omega}\Big{(}c_{F}\chi-
c_{F}+2(\psi_{\varepsilon}-c_{F})^{-}\Big{)}\eta\circ\mathrm{d}_{\Sigma}\,dx+\int_{\Omega}\left(\psi_{\varepsilon}-c_{F}\right)^{+}|\eta\circ\mathrm{d}_{\Sigma}|\,dx.$
(1.14)

In (1.14)
$\chi(\cdot,t)=\mathbf{1}_{\Omega_{t}^{+}}-\mathbf{1}_{\Omega_{t}^{-}}$ and
$g^{\pm}$ denotes the positive/negative parts of a function $g$ respectively,
and $\eta$ is a $\delta_{0}$-truncation of the identity function (cf. (3.8)).
In particular, we have $\eta\circ\mathrm{d}_{\Sigma}\geq 0$ in $\Omega$ due to
our convention on the signed-distance function, and thus the integrands in
(1.14) are all non-negative. See the proof of Theorem 3.2 below for more
details.

The main result of this work is the following:

######  Theorem 1.1.

Assume that the surface $\Sigma_{t}$ (1.8) evolves by mean curvature flow
during $[0,T]$. If the initial datum of (1.7) is well-prepared in the sense
that

$\varepsilon\|\mathbf{u}_{\varepsilon}(\cdot,0)\|_{L^{\infty}}+B[\mathbf{u}_{\varepsilon}|\Sigma](0)+E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](0)\leq
C_{1}\varepsilon$ (1.15)

for some constant $C_{1}$ that is independent of $\varepsilon$, then there
exists $C_{2}$ independent of $\varepsilon$ so that

$\displaystyle\sup_{t\in[0,T]}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](t)\leq
C_{2}\varepsilon,$ (1.16)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}|\psi_{\varepsilon}-c_{F}\mathbf{1}_{\Omega_{t}^{+}}|\,dx\leq
C_{2}\varepsilon^{1/2},$ (1.17) $\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{F\left(\mathbf{u}_{\varepsilon}\right)}{\varepsilon}\right)\,dx=c_{F}\mathcal{H}^{d-1}(\Sigma_{t}),\quad\forall
t\in[0,T].$ (1.18)

Moreover, for some subsequence $\varepsilon_{k}\downarrow 0$ there holds

$\mathbf{u}_{\varepsilon_{k}}\xrightarrow{k\to\infty}\mathbf{u}^{\pm}~{}\text{weakly
in}~{}L^{2}(0,T;H^{1}_{loc}(\Omega^{\pm}_{t})),$ (1.19)

where $\mathbf{u}^{\pm}$ are weak solutions to the harmonic heat flows into
$\mathfrak{m}_{\pm}$ respectively and

$\mathbf{u}^{\pm}\in
L^{\infty}\left(0,T;H^{1}(\Omega^{\pm}_{t};\mathfrak{m}_{\pm})\right),\quad\partial_{t}\mathbf{u}^{\pm}\in
L^{2}\left(0,T;L^{2}_{loc}(\Omega^{\pm}_{t})\right).$ (1.20)

Furthermore, for almost every $t\in(0,T)$,
$(\mathbf{u}^{+},\mathbf{u}^{-})|_{\Sigma_{t}}$ is a minimal pair, i.e.

$|\mathbf{u}^{+}-\mathbf{u}^{-}|_{\mathbb{R}^{n}}(x,t)=\mathrm{dist}_{\mathfrak{m}}\qquad\mathcal{H}^{d-1}-a.e.~{}x\in\Sigma_{t}.$
(1.21)

A few comments are in order. Firstly, in (1.15) the $L^{\infty}$ bound of the
initial datum is used (together with (1.6)) to obtain an uniform in space-time
$L^{\infty}$-bound of $\mathbf{u}_{\varepsilon}$, i.e.

$\|\mathbf{u}_{\varepsilon}\|_{L^{\infty}(\Omega\times(0,T))}\leq c_{0}$
(1.22)

for some fixed constant $c_{0}$. Such an estimate, derived by applying the
maximum principle to (1.7a), enables us to avoid several technical
complications in the passage of the limit $\varepsilon\downarrow 0$. Indeed,
even in the case when $d=2$, severe difficulties arise in the anisotropic
model considered in [26] where an estimate like (1.22) is not available.
Secondly, we also get a bound like
$\sup_{t\in[0,T]}B[\mathbf{u}_{\varepsilon}|\Sigma](t)\leq C\varepsilon$.
However, we merely use such an estimate to derive (1.17), which is crucial to
obtain the minimal pair condition (1.21). Now we turn to the discussion of the
limiting maps $\mathbf{u}^{\pm}$. If we denote the second fundamental forms of
$\mathfrak{m}_{\pm}$ at points $\mathbf{p}^{\pm}$ by
$A^{\pm}(\mathbf{p}^{\pm})(\cdot,\cdot)$, respectively. Then the theorem above
claims that the pair of mappings

$\mathbf{u}^{\pm}(\cdot,t):\Omega^{\pm}_{t}\mapsto\mathfrak{m}_{\pm}\subset\mathbb{R}^{n}$
(1.23)

satisfies the following system in the weak sense:

$\displaystyle\partial_{t}\mathbf{u}^{\pm}-\Delta\mathbf{u}^{\pm}$
$\displaystyle=A^{\pm}(\mathbf{u}^{\pm})(\nabla\mathbf{u}^{\pm},\nabla\mathbf{u}^{\pm})$
$\displaystyle~{}\text{in}~{}\cup_{t\in[0,T]}\Omega^{\pm}_{t}\times\\{t\\},$
(1.24a) $\displaystyle|\mathbf{u}^{+}-\mathbf{u}^{-}|_{\mathbb{R}^{n}}$
$\displaystyle=\mathrm{dist}_{\mathfrak{m}}$
$\displaystyle~{}\mathcal{H}^{d-1}\text{- a.e on }~{}\Sigma_{t},$ (1.24b)
$\displaystyle\mathbf{u}^{-}$ $\displaystyle=\mathbf{g},$
$\displaystyle~{}\text{on}~{}\partial\Omega.$ (1.24c)

The equation (1.24a) say $\mathbf{u}^{\pm}$ are harmonic map heat flows from
the moving domain $\Omega^{\pm}_{t}$ to the target manifold
$\mathfrak{m}_{\pm}$. The equation (1.24b) is referred to as the minimal pair
boundary condition by [24, 25]. In (1.24c),
$\mathbf{g}:\overline{\Omega}\mapsto\mathfrak{m}_{-}$ is a smooth mapping.

To make the main theorem applicable, we shall show that the class of initial
datum fulfilling the condition (1.15) is geometrically rich. This is stated in
the following result.

######  Theorem 1.2.

For any $\delta\in(0,\delta_{0})$ and any pair of mappings
$\mathbf{u}^{in}_{\pm}\in H^{1}(\Omega_{0}^{\pm},\mathfrak{m}_{\pm})$ with

$|\mathbf{u}^{in}_{+}(p)-\mathbf{u}^{in}_{-}(p)|_{\mathbb{R}^{n}}=\mathrm{dist}_{\mathfrak{m}},\quad\forall
p\in\Sigma_{0},$ (1.25)

there exist $\mathbf{u}^{in}_{\varepsilon}\in H^{1}(\Omega,\mathbb{R}^{n})\cap
L^{\infty}(\Omega)$ and a constant $C=C(\delta,\mathbf{u}^{in}_{\pm})$ so that

$\displaystyle\mathbf{u}_{\varepsilon}^{in}$
$\displaystyle=\mathbf{u}^{in}_{\pm}~{}\text{ inside
}~{}\Omega_{0}^{\pm}\backslash B_{2\delta}(\Sigma_{0}),$ (1.26) $\displaystyle
E_{\varepsilon}[\mathbf{u}_{\varepsilon}^{in}|\Sigma_{0}]$ $\displaystyle\leq
C\varepsilon,$ (1.27) $\displaystyle
B[\mathbf{u}_{\varepsilon}^{in}|\Sigma_{0}]$ $\displaystyle\leq C\varepsilon.$
(1.28)

The rest of the work will be organized as follows: in Section 2, we shall
recall results that will be employed throughout the work. These include the
compactness and closure of special function with bounded variation (cf. [3,
Chapter 4]), the theory of minimal connection developed by Sternberg [36] and
Lin–Pan–Wang [24], the elements of differential geometry used in the
description of interface motion, and finally the relative entropy method by
Fischer–Laux–Simon [16]. In particular, in Subsection 2.5, we shall adapt this
later method to system (1.7), and then derive a differential inequality, i.e.
Proposition 2.10. Such an inequality was first derived by Laux–Liu [21] for a
matrix-valued system. This Proposition, when combined with Chen–Struwe [8]
along with other results in Section 3, leads to the convergences to harmonic
heat flows locally away from $\Sigma_{t}$. Another important consequence of
Proposition 2.10 is a sharp $L^{1}$-convergence rate estimate of
$\psi_{\varepsilon}$, obtained in Theorem 3.2. This theorem will be used to
derive fine estimates of the level sets of $\psi_{\varepsilon}$ in Lemma 3.4,
as well as convergences of some corrections of $\mathbf{u}_{\varepsilon}$ up
to the free boundary $\Sigma_{t}$. All of these will be done in Section 3, and
we shall use them to derive the minimal pair boundary condition (1.21) in
Section 4, and thus finish the proof of Theorem 1.1. Finally we prove Theorem
1.2 in Section 5.

We end the introductory part by introducing the notations and conventions that
will be employ through this work. Unless specified otherwise $C>0$ is a
generic constant whose value might change from line to line, and will depend
only on the geometry of the interface (1.8) but not on $\varepsilon$ or
$t\in[0,T]$. In order to simplify the presentation, we shall sometimes
abbreviate the estimates like $X\leq CY$ by $X\lesssim Y$ for some non-
negative quantities $X,Y$.

We provide a list of symbols for the convenience of the readers:

  * •

$A:B$ is the Frobenius inner product of two square matrices $A,B$, defined by
$\operatorname{tr}A^{\mathsf{T}}B$.

  * •

$\partial_{i}=\partial_{x_{i}}~{}(0\leq i\leq d)$ with the convention that
$\partial_{t}=\partial_{0}$.

  * •

$\nabla f$ is the (distributional) gradient of a function $f$ with variables
$x=(x_{1},\cdots,x_{d})$.

  * •

$\partial W=(\partial_{u_{1}}W,\cdots,\partial_{u_{n}}W)$ is the gradient of a
smooth function $W=W(\mathbf{u})$.

  * •

$\partial\mathrm{d}_{F}(\mathbf{u})$: the generalized gradient of
$\mathrm{d}_{F}$ (cf. (2.37)).

  * •

$\partial U$: measure-theoretic boundary of a set $U$ of finite perimeter with
measure-theoretic outer normal vector $\nu$.

  * •

$\mathrm{dist}(\mathbf{u},A)$: distance from $\mathbf{u}$ to
$A\subset\mathbb{R}^{n}$.

  * •

$\mathrm{dist}_{\mathfrak{m}}$: the distance between $\mathfrak{m}_{\pm}$ in
$\mathbb{R}^{n}$, i.e.
$\mathrm{dist}_{\mathfrak{m}}:=\inf_{\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm}}\left|\mathbf{p}^{+}-\mathbf{p}^{-}\right|_{\mathbb{R}^{n}}$.

  * •

$\mathrm{d}_{\mathfrak{m}}(\mathbf{u})$: signed-distance from
$\mathbf{u}\in\mathbb{R}^{n}$ to
$\mathfrak{m}=\mathfrak{m}_{+}\sqcup\mathfrak{m}_{-}$ (cf. (2.17)).

  * •

$U_{\pm}$ are the domains enclosed by $\mathfrak{m}_{\pm}$ respectively, and
$U_{0}=\mathbb{R}^{n}-\overline{U_{+}\cup U_{-}}$ (cf. (2.15)).

  * •

$B_{\delta}(U)$: the $\delta$-(tubular) neighborhood of a set $U$ in the
corresponding Euclidean space. In particular, $B_{\delta}(x)$ is the ball
centered at $x$.

##  2\. Preliminaries

###  2.1. Special function of bounded variation

######  Definition 2.1.

We say that $\mathbf{u}\in BV(\Omega,\mathbb{R}^{n})$ is a special function
with bounded variation and we write $\mathbf{u}\in SBV(\Omega)$, if the Cantor
part of its distributional derivative $\nabla^{c}u$ vanishes, i.e.

$\nabla\mathbf{u}=\nabla^{a}\mathbf{u}\,\mathcal{L}^{d}+\left(\mathbf{u}^{+}-\mathbf{u}^{-}\right)\otimes\nu_{\mathbf{u}}~{}\mathcal{H}^{d-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}J_{\mathbf{u}}\quad\forall\mathbf{u}\in
SBV(\Omega)$ (2.1)

where $\nabla^{a}$ denotes the absolutely continuous part of the
distributional derivative (with respect to Lebesgue measure $\mathcal{L}^{d}$)
and $J_{\mathbf{u}}$ is the jump set of $\mathbf{u}$ with measure theoretical
outer normal vector $\nu_{\mathbf{u}}$.

The following two results will be used to obtain convergences up to the free
boundary. We refer the monograph of Ambrosio–Fusco–Pallara [3] for proofs.

######  Proposition 2.2.

(Closure of $SBV$) Let $\varphi:[0,\infty)\rightarrow[0,\infty],$ be lower
semicontinuous increasing functions and assume that
$\lim_{t\rightarrow\infty}\frac{\varphi(t)}{t}=\infty$. Let
$\Omega\subset\mathbb{R}^{n}$ be open and bounded, and let
$\\{\mathbf{u}_{k}\\}\subset SBV(\Omega)$ such that

$\sup_{k}\int_{\Omega}\varphi\left(\left|\nabla^{a}\mathbf{u}_{k}\right|\right)dx+\sup_{k}\int_{J_{\mathbf{u}_{k}}}\left|\mathbf{u}_{k}^{+}-\mathbf{u}_{k}^{-}\right|\,d\mathcal{H}^{d-1}<\infty$
(2.2)

If $\\{\mathbf{u}_{k}\\}$ weakly-star converges in $BV(\Omega)$ to
$\mathbf{u}$, then $\mathbf{u}\in SBV(\Omega)$, the absolute continuous part
of the gradients $\nabla^{a}\mathbf{u}_{k}$ weakly converge to
$\nabla^{a}\mathbf{u}$ in $L^{1}(\Omega)$, and the jump part of the gradient
$\nabla^{j}\mathbf{u}_{k}$ weakly-star converge to $\nabla^{j}\mathbf{u}$ in
$\Omega$. Moreover,

$\displaystyle\int_{\Omega}\varphi(|\nabla^{a}\mathbf{u}|)dx\leq\liminf_{k\rightarrow\infty}\int_{\Omega}\varphi\left(\left|\nabla^{a}\mathbf{u}_{k}\right|\right)dx\quad\text{
if }\varphi\text{ is convex }.$ (2.3)

######  Proposition 2.3.

(Compactness of $SBV$) Let $\varphi,\Omega$ as in Proposition 2.2. Let
$\\{\mathbf{u}_{k}\\}\subset SBV(\Omega)$ be satisfying (2.2) and assume, in
addition, that $\left\|\mathbf{u}_{k}\right\|_{L^{\infty}}$ is uniformly
bounded in $k$. Then, there exists a subsequence

$\mathbf{u}_{k}\xrightarrow{k\to\infty}\mathbf{u}\in SBV(\Omega)\text{ weakly
star in $BV(\Omega)$}.$ (2.4)

###  2.2. Minimal connections

Now let’s briefly describe some basic properties of minimal orbits. For any
$\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm}$, we define their minimal connection by

$\displaystyle\mathcal{C}_{F}(\mathbf{p}^{+},\mathbf{p}^{-})$
$\displaystyle:=\inf\left\\{\int_{\mathbb{R}}\frac{1}{2}|\bm{\gamma}^{\prime}(t)|^{2}+F(\bm{\gamma}(t))\,dt\,\Big{|}\,\bm{\gamma}\in
H^{1}(\mathbb{R},\mathbb{R}^{n}),\bm{\gamma}(\pm\infty)=\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm}\right\\}.$
(2.5)

######  Lemma 2.4.

The function
$\mathcal{C}_{F}(\mathbf{p}^{+},\mathbf{p}^{-}):\mathfrak{m}_{+}\times\mathfrak{m}_{-}\mapsto\mathbb{R}^{+}$
is Lipschitz continuous. Moreover, Let $c_{F}$ be the number defined by
(1.13), then

$c_{F}=\inf_{\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm}}\mathcal{C}_{F}(\mathbf{p}^{+},\mathbf{p}^{-}),$
(2.6)

To proceed, we define the centralized potential, the even function

$\widetilde{F}(\lambda):=\begin{cases}f\left(\left(\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}+\lambda\right)^{2}\right)&\text{
if }\lambda\leq 0,\\\
f\left(\left(\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}-\lambda\right)^{2}\right)&\text{
if }\lambda\geq 0,\end{cases}$ (2.7)

and the associated scalar-valued minimal connection problem

$c_{\widetilde{F}}:=\min\left\\{\int_{\mathbb{R}}\left(\frac{1}{2}\left|\bm{\gamma}^{\prime}(s)\right|^{2}+\widetilde{F}(\bm{\gamma}(s))\right)dt~{}\Big{|}~{}\bm{\gamma}\in
H^{1}(\mathbb{R}),\bm{\gamma}(\pm\infty)=\pm\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\right\\}.$
(2.8)

######  Lemma 2.5.

It holds that

$c_{\widetilde{F}}=2\int_{0}^{\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}d\lambda\quad=2\int_{0}^{\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2f\left(\lambda^{2}\right)}d\lambda.$
(2.9)

There exists a minimizer $\alpha(s)$ of (2.8) that also satisfies

$\displaystyle\alpha(s)\in
C^{\infty}\left(\mathbb{R},\left(-\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\right)\right)\text{
is odd and strictly increasing in }\mathbb{R},$ (2.10a)
$\displaystyle-\alpha^{\prime\prime}(s)+\widetilde{F}^{\prime}(\alpha(s))=0,\quad
s\in\mathbb{R};\quad\alpha(\pm\infty)=\pm\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}.$
(2.10b)
$\displaystyle\alpha^{\prime}(s)=\sqrt{2\widetilde{F}(\alpha(s))}\quad\forall
s\in\mathbb{R},$ (2.10c)
$\displaystyle\left|\alpha^{\prime}(s)\right|+\left|\alpha(s)\mp\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\right|\leq
Ce^{-C|s|}\quad\text{ as }s\rightarrow\pm\infty.$ (2.10d)

We need an equivalent condition to the minimal pair one stated at (1.21). To
this end, we introduce

$\displaystyle
M^{+}:=\left\\{\mathbf{p}^{+}\in\mathfrak{m}_{+}:\exists\,\mathbf{p}^{-}\in\mathfrak{m}_{-}\text{
s.t.
}\left|\mathbf{p}^{+}-\mathbf{p}^{-}\right|=\mathrm{dist}_{\mathfrak{m}}\right\\},$
(2.11) $\displaystyle
M^{-}:=\left\\{\mathbf{p}^{-}\in\mathfrak{m}_{-}:\exists\,\mathbf{p}^{+}\in\mathfrak{m}_{+}\text{
s.t.
}\left|\mathbf{p}^{+}-\mathbf{p}^{-}\right|=\mathrm{dist}_{\mathfrak{m}}\right\\}$
(2.12)

######  Lemma 2.6.

[24, Theorem 2.1] The function
$\mathcal{C}_{F}(\cdot,\cdot):\mathfrak{m}_{+}\times\mathfrak{m}_{-}\mapsto[0,\infty)$
is Lipschitz continuous, and
$\mathcal{C}_{F}(\mathbf{p}^{+},\mathbf{p}^{-})\geq c_{F}$ for any
$\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm}$. Moreover, we have the equivalence

$\displaystyle\mathbf{p}^{\pm}\in\mathfrak{m}_{\pm},\quad\mathcal{C}_{F}(\mathbf{p}^{+},\mathbf{p}^{-})=c_{F}\Longleftrightarrow\mathbf{p}^{\pm}\in
M^{\pm},\quad|\mathbf{p}^{+}-\mathbf{p}^{-}|=\mathrm{dist}_{\mathfrak{m}}.$
(2.13)

Furthermore, assuming the left-hand side of (2.13). Then the corresponding
minimal connecting orbit $\bm{\gamma}\in H^{1}(\mathbb{R},\mathbb{R}^{k})$
attaining $c^{F}$ (2.8) is the line segment

$\bm{\gamma}(t)=\tfrac{\mathbf{p}^{+}+\mathbf{p}^{-}}{2}+\alpha(t)\tfrac{\mathbf{p}^{+}-\mathbf{p}^{-}}{\mathrm{dist}_{\mathfrak{m}}},\quad
t\in\mathbb{R},$ (2.14)

where $\alpha\in H^{1}(\mathbb{R})$ is a solution to (2.8) and equivalently
(2.10b).

Seemly more complicated, the condition on the LHS of (2.13) is more compatible
to the variational structure of the functional (1.2) compared with the one on
the RHS.

###  2.3. Quasi-distance function

To proceed, we introduce the quasi-distance function. To define it, we denote
by

$\left\\{\begin{split}&U_{\pm}\text{ the bounded open domain enclosed by
}\mathfrak{m}_{\pm},\\\ &U_{+}\cap U_{-}=\emptyset,\qquad
U_{0}=\mathbb{R}^{n}-\overline{U_{+}\cup U_{-}}.\end{split}\right.$ (2.15)

We denote the signed-distance function from $\mathbf{u}\in\mathbb{R}^{n}$ to
the set $\mathfrak{m}_{\pm}$ by $\mathrm{d}_{\mathfrak{m}_{\pm}}(\mathbf{u})$,
and make the following sign convention so that
$\mathrm{d}_{\mathfrak{m}_{\pm}}$ are smooth in
$B_{\delta_{0}}(\mathfrak{m}_{\pm})$:

$\mathrm{d}_{\mathfrak{m}_{\pm}}(\mathbf{u})>0\text{ for }\mathbf{u}\in
U_{\pm}.$ (2.16)

Then we define the signed-distance from $\mathbf{u}$ to $\mathfrak{m}$
according to its distance to its two components:

$\mathrm{d}_{\mathfrak{m}}(\mathbf{u})=\left\\{\begin{split}\mathrm{d}_{\mathfrak{m}_{+}}(\mathbf{u})&\qquad\text{
when
}\mathrm{dist}(\mathbf{u},\mathfrak{m}_{+})\leq\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\\\
\mathrm{d}_{\mathfrak{m}_{-}}(\mathbf{u})&\qquad\text{ when
}\mathrm{dist}(\mathbf{u},\mathfrak{m}_{-})\leq\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}.\end{split}\right.$
(2.17)

With these notations, we define

$\mathrm{d}_{F}(\mathbf{u}):=\left\\{\begin{split}\tfrac{1}{2}c_{F}\quad&\text{
if
}\mathrm{dist}(\mathbf{u},\mathfrak{m}_{-})>\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\mathbf{u}\in
U_{-},\\\
\int_{0}^{|\mathrm{d}_{\mathfrak{m}}(\mathbf{u})|}\sqrt{2f(\lambda^{2})}\,d\lambda\quad&\text{
if
}\mathrm{dist}(\mathbf{u},\mathfrak{m}_{-})\leq\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\\\
\tfrac{1}{2}c_{F}\quad&\text{ if
}\mathrm{dist}(\mathbf{u},\mathfrak{m})>\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\mathbf{u}\in
U_{0},\\\
c_{F}-\int_{0}^{\mathrm{d}_{\mathfrak{m}}(\mathbf{u})}\sqrt{2f(\lambda^{2})}\,d\lambda\quad&\text{
if
}\mathrm{dist}(\mathbf{u},\mathfrak{m}_{+})\leq\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\mathbf{u}\in
U_{0};\text{ or }\mathbf{u}\in U_{+}\end{split}\right.$ (2.18)

where $c_{F}$ is the surface tension coefficient (1.13). The function (2.18)
is a modification of the one used in [36, 17]. Important properties of (2.18)
is summarized in the following lemma.

######  Lemma 2.7.

The function $\mathrm{d}_{F}(\mathbf{u})$ (defined by (2.18)) is $C^{1}$ in
$B_{\delta_{0}}(\mathfrak{m})$ and
$\frac{\partial\mathrm{d}_{F}(\mathbf{u})}{|\partial\mathrm{d}_{F}(\mathbf{u})|}$
is a continuous unit vector field in $B_{\delta_{0}}$ so that

$\frac{\partial\mathrm{d}_{F}(\mathbf{u})}{|\partial\mathrm{d}_{F}(\mathbf{u})|}=\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u})\qquad\forall\mathbf{u}\in
B_{\delta_{0}}(\mathfrak{m}).$ (2.19)

Moreover, it is Lipschitz continuous in $\mathbb{R}^{n}$, and

$|\partial\mathrm{d}_{F}(\mathbf{u})|\leq\sqrt{2F(\mathbf{u})}\quad
a.e.~{}\mathbf{u}\in\mathbb{R}^{n},$ (2.20)

and is related to $c_{F}$ (2.8) by

$\displaystyle\mathrm{d}_{F}(\mathbf{u})$
$\displaystyle=\left\\{\begin{array}[]{rl}0\qquad\text{if and only
if}&~{}\mathbf{u}\in\mathfrak{m}_{-},\\\ c_{F}\qquad\text{if and only
if}&~{}\mathbf{u}\in\mathfrak{m}_{+}.\end{array}\right.$ (2.23)

###### Proof.

The continuity of $\mathrm{d}_{F}$ follows from (1.13). To show
$\mathrm{d}_{F}\in C^{1}(B_{\delta_{0}}(\mathfrak{m}))$, it suffices to look
into the second and the fourth cases in its definition (2.18). In
$B_{\delta_{0}}(\mathfrak{m})$, the possible singularities of its derivative
are points in $\mathfrak{m}_{-}$, which are removable. Indeed, using the last
condition in (1.6), one can verify that

$h(s)=\int_{0}^{s^{1/2}}\sqrt{2f(\lambda^{2})}\,d\lambda\in C^{1}[0,\infty).$
(2.24)

As a result, we have from (2.18) that
$\mathrm{d}_{F}(\mathbf{u})=h(\mathrm{d}_{\mathfrak{m}}^{2})$ in
$B_{\delta_{0}}(\mathfrak{m}_{-})$. This also implies (2.19) by taking one-
sided limits approaching $\mathfrak{m}$. We note that the signed distance
function $\mathrm{d}_{\mathfrak{m}}$ is smooth in
$B_{\delta_{0}}(\mathfrak{m})$.

It is obvious that $\mathrm{d}_{F}$ is Lipschitz in each subdomain where it is
defined. So it suffices to check the Lipschitz condition across adjacent
regions. For instance, if
$\mathrm{dist}(\mathbf{u}_{1},\mathfrak{m}_{-})>\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2},\mathbf{u}_{1}\in
U_{-}$ and
$\mathrm{dist}(\mathbf{u}_{2},\mathfrak{m}_{-})\leq\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}$,
then by (1.13) and the first two cases in (2.18),

$\displaystyle 0$
$\displaystyle\leq\mathrm{d}_{F}(\mathbf{u}_{1})-\mathrm{d}_{F}(\mathbf{u}_{2})=\int_{\mathrm{dist}(\mathbf{u}_{2},\mathfrak{m}_{-})}^{\frac{1}{2}\mathrm{dist}_{\mathfrak{m}}}\sqrt{2f(\lambda^{2})}\,d\lambda$
$\displaystyle\leq\int_{\mathrm{dist}(\mathbf{u}_{2},\mathfrak{m}_{-})}^{\mathrm{dist}(\mathbf{u}_{1},\mathfrak{m}_{-})}\sqrt{2f(\lambda^{2})}\,d\lambda\leq
C\left(\mathrm{dist}(\mathbf{u}_{2},\mathfrak{m}_{-})-\mathrm{dist}(\mathbf{u}_{1},\mathfrak{m}_{-})\right)\leq
C|\mathbf{u}_{2}-\mathbf{u}_{1}|.$

Other cases can be treated in a similar way. The inequality (2.20) and the
formula (2.23) follows directly from the definition (2.18). ∎

###  2.4. Geometry of interfaces

Under a local parametrization $\bm{\varphi}_{t}(s):U\to\Sigma_{t}$, the MCF
writes

$\partial_{t}\bm{\varphi}_{t}(s)\cdot\mathbf{n}(s,t)=\mathbf{H}(\bm{\varphi}_{t}(s),t)\cdot\mathbf{n}(s,t)$
(2.25)

where $\mathbf{H}$ is the (mean) curvature vector pointing to the inner normal
$\mathbf{n}$. For $\delta>0$, the $\delta$-neighborhood of $\Sigma_{t}$ are
the open sets

$B_{\delta}(\Sigma_{t}):=\\{x\in\Omega:|\mathrm{d}_{\Sigma}(x,t)|<\delta\\}.$
(2.26)

We shall choose the $\delta_{0}$ (first appeared in (1.6)) smaller enough so
that the nearest point projection

$P_{\Sigma}(\cdot,t):B_{4\delta_{0}}(\Sigma_{t})\mapsto\Sigma_{t}$

is smooth for any $t\in[0,T]$, and the interface (1.8) stays at least
$\delta_{0}$ distance away from the boundary of the domain $\partial\Omega$.
Analytically we have

$P_{\Sigma}(x,t)=x-\nabla\mathrm{d}_{\Sigma}(x,t)\mathrm{d}_{\Sigma}(x,t).$

So for each fixed $t\in[0,T]$, any point $x\in B_{4\delta_{0}}(\Sigma_{t})$
corresponds to a unique pair $(r,s)$ with $r=\mathrm{d}_{\Sigma}(x,t)$ and
$s\in\mathbb{T}^{1}$, and thus the identity

$\mathrm{d}_{\Sigma}(\bm{\varphi}_{t}(s)+r\mathbf{n}(s,t),t)\equiv r$

holds with independent variables $(r,s,t)$. Differentiating this identity with
respect to $r$ and $t$ leads to the following identities:

$\nabla\mathrm{d}_{\Sigma}(x,t)=\mathbf{n}(s,t),\qquad-\partial_{t}\mathrm{d}_{\Sigma}(x,t)=\partial_{t}\bm{\varphi}_{t}(s)\cdot\mathbf{n}(s,t)=:V(s,t).$
(2.27)

This extends the normal vector and the normal velocity of $\Sigma_{t}$ to a
neighborhood of it. Recall in (2.28) that we extend the normal vector field
$\mathbf{n}$ of the interface $\Sigma_{t}$ to a neighborhood of it.

Now we come to the definition of $\bm{\xi}$ in the modulated energy
$E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](t)$ (1.11). This is done by
extending the inner normal vector field $\mathbf{n}$ through

$\bm{\xi}(x,t)=\phi\left(\frac{\mathrm{d}_{\Sigma}(x,t)}{\delta_{0}}\right)\nabla\mathrm{d}_{\Sigma}(x,t).$
(2.28)

In (2.28) $\phi(x)\geq 0$ is an even, smooth function on $\mathbb{R}$ that
decreases for $x\in[0,1]$, and satisfies

$\begin{cases}\phi(x)>0~{}&\text{ for }~{}|x|<1,\\\ \phi(x)=0~{}&\text{ for
}~{}|x|\geq 1,\\\ 1-4x^{2}\leq\phi(x)\leq 1-\frac{1}{2}x^{2}~{}&\text{ for
}~{}|x|\leq 1/2.\end{cases}$ (2.29)

To fulfill these requirements, we can simply choose

$\phi(x)=e^{\frac{1}{x^{2}-1}+1}~{}\text{for}~{}|x|<1~{}\text{and}~{}\phi(x)=0~{}\text{for}~{}|x|\geq
1.$ (2.30)

We also need to extend the curvature of (1.8). To proceed, choose a cut-off
function

$\eta_{0}\in C_{c}^{\infty}(B_{2\delta_{0}}(\Sigma_{t}))~{}\text{ with
}\eta_{0}=1\text{ in }B_{\delta_{0}}(\Sigma_{t}).$ (2.31)

We extend the curvature vector $\mathbf{H}$ by

$\mathbf{H}(x,t)=\kappa\nabla\mathrm{d}_{\Sigma}(x,t)\quad\text{with}\quad\kappa(x,t)=-\Delta\mathrm{d}_{\Sigma}(P_{\Sigma}(x,t))\eta_{0}(x,t).$
(2.32)

By (2.31), $\mathbf{H}$ is extended constantly in the normal direction. So we
have

$\displaystyle(\mathbf{n}\cdot\nabla)\mathbf{H}$ $\displaystyle=0\text{ and
}(\bm{\xi}\cdot\nabla)\mathbf{H}=0\quad\forall t\in[0,T],~{}x\in
B_{\delta_{0}}(\Sigma_{t}).$ (2.33)

Moreover, by (2.28) we have

$\bm{\xi}=0~{}\text{and}~{}\mathbf{H}=0\quad\forall
t\in[0,T],~{}x\in\partial\Omega.$ (2.34)

###  2.5. Modulated energy method

As the gradient flow of the Ginzburg–Landau energy (1.2), the system (1.7a)
has the following energy dissipation law

$A_{\varepsilon}(\mathbf{u}_{\varepsilon}(\cdot,T))+\int_{0}^{T}\int_{\Omega}\varepsilon|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dx\,dt=A_{\varepsilon}(\mathbf{u}_{\varepsilon}(\cdot,0)),~{}\text{for
all}~{}T>0.$ (2.35)

For initial data undergoing a transition near the interface $\Sigma_{t}$, due
to concentrations of $\nabla\mathbf{u}_{\varepsilon}$ near $\Sigma_{t}$, the
dissipation law (2.35) is not sufficient to derive quantitative convergences
of $\mathbf{u}_{\varepsilon}$, even away from $\Sigma_{t}$. Following a recent
work of Fisher et al. [16] we shall develop in this section a calibrated
inequality, which modulates the concentration and obtain the compactness of
$\\{\mathbf{u}_{\varepsilon}\\}$ in Sobolev spaces.

Recalling the notions in Subsection 2.4, we claim the following identities
which will be employed to prove the calibrated inequality:

$\displaystyle\nabla\cdot\bm{\xi}+\mathbf{H}\cdot\bm{\xi}$
$\displaystyle=O(\mathrm{d}_{\Sigma}),$ (2.36a)
$\displaystyle\partial_{t}\mathrm{d}_{\Sigma}(x,t)+(\mathbf{H}(x,t)\cdot\nabla)\mathrm{d}_{\Sigma}(x,t)$
$\displaystyle=0\quad\text{in}~{}B_{\delta_{0}}(\Sigma_{t}).$ (2.36b)
$\displaystyle\partial_{t}\bm{\xi}+\left(\mathbf{H}\cdot\nabla\right)\bm{\xi}+\left(\nabla\mathbf{H}\right)^{\mathsf{T}}\bm{\xi}$
$\displaystyle=0\quad\text{in}~{}B_{\delta_{0}}(\Sigma_{t}),$ (2.36c)
$\displaystyle\partial_{t}|\bm{\xi}|^{2}+\left(\mathbf{H}\cdot\nabla\right)|\bm{\xi}|^{2}$
$\displaystyle=0\quad\text{in}~{}B_{\delta_{0}}(\Sigma_{t}),$ (2.36d)

where $\nabla\mathbf{H}:=\\{\partial_{j}H_{i}\\}_{1\leq i,j\leq d}$ is a
matrix with $i$ being the row index.

###### Proof of (2.36).

Recalling (2.28), $\phi_{0}(\tau):=\phi(\frac{\tau}{\delta_{0}})$ is an even
function. So it follows from $\phi_{0}^{\prime}(0)=0$ and Taylor’s expansion
in $\mathrm{d}_{\Sigma}$ that

$\displaystyle\nabla\cdot\bm{\xi}$
$\displaystyle=|\nabla\mathrm{d}_{\Sigma}|^{2}\phi_{0}^{\prime}(\mathrm{d}_{\Sigma})+\phi_{0}(\mathrm{d}_{\Sigma})\Delta\mathrm{d}_{\Sigma}(x,t)$
$\displaystyle=O(\mathrm{d}_{\Sigma})+\phi_{0}(\mathrm{d}_{\Sigma})\Delta\mathrm{d}_{\Sigma}(P_{I}(x,t),t),$

and this together with (2.32) leads to (2.36a). Using (2.27) and (2.32), we
can write (2.25) as the transport equation (2.36b). By (2.36b) we have the
following identities in $B_{\delta_{0}}(\Sigma_{t})$:

$\displaystyle\partial_{t}\nabla\mathrm{d}_{\Sigma}+(\mathbf{H}\cdot\nabla)\nabla\mathrm{d}_{\Sigma}+(\nabla\mathbf{H})^{\mathsf{T}}\nabla\mathrm{d}_{\Sigma}=0,$
$\displaystyle\partial_{t}\phi_{0}(\mathrm{d}_{\Sigma})+(\mathbf{H}\cdot\nabla)\phi_{0}(\mathrm{d}_{\Sigma})=0.$

These two equations together imply (2.36c). Finally (2.36d) is a consequence
of (2.36c) and (2.33). ∎

Now we discuss the differentiability of $\psi_{\varepsilon}$ (cf. (1.12)). It
follows from Lemma 2.7 that $\mathrm{d}_{F}(\cdot)$ is a Lipschitz function in
$\mathbb{R}^{n}$ with $\mathrm{d}_{F}(0)=0$ under the assumption that
$0\in\mathfrak{m}^{-}$. Following Laux–Simon [22], for every
$(x,t)\in\Omega\times[0,T]$, we consider the restriction of
$\mathrm{d}_{F}(\mathbf{u}_{\varepsilon}(x,t))$ to the affine space

$\displaystyle
T^{\mathbf{u}_{\varepsilon}}_{x,t}:=\mathbf{u}_{\varepsilon}(x,t)+{\rm
span}\\{\partial_{0}\mathbf{u}_{\varepsilon}(x,t),\cdots,\partial_{d}\mathbf{u}_{\varepsilon}(x,t)\\},$

denoted by $\mathrm{d}_{F}|_{T^{\mathbf{u}_{\varepsilon}}_{x,t}}$. By the
generalized chain rule [2],
$\mathrm{d}_{F}|_{T^{\mathbf{u}_{\varepsilon}}_{x,t}}$ is differentiable at
$\mathbf{u}_{\varepsilon}(x,t)$. Now we denote the orthogonal projection from
$\mathbb{R}^{n}$ to the subspace
$T^{\mathbf{u}_{\varepsilon}}_{x,t}-\mathbf{u}_{\varepsilon}(x,t)$ by
$\Pi_{x,t}^{\mathbf{u}_{\varepsilon}}$, and define the generalized
differential by

$\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon}):=\partial\left(\mathrm{d}_{F}|_{T^{\mathbf{u}_{\varepsilon}}_{x,t}}\right)\Big{|}_{\mathbf{u}_{\varepsilon}(x,t)}\circ\Pi^{\mathbf{u}_{\varepsilon}}_{x,t}.$
(2.37)

Then we have for $0\leq i\leq d$ that

$\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{i}\mathbf{u}_{\varepsilon}=\partial\left(\mathrm{d}_{F}|_{T^{\mathbf{u}_{\varepsilon}}_{x,t}}\right)\Big{|}_{\mathbf{u}_{\varepsilon}(x,t)}\cdot\partial_{i}\mathbf{u}_{\varepsilon}=\partial_{i}(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon})$
(2.38)

where the second equality is due to the directional derivative at
$\mathbf{u}_{\varepsilon}(x,t)$ pointing to
$\partial_{i}\mathbf{u}_{\varepsilon}(x,t)$. This proves the generalized chain
rule

$\displaystyle\partial_{i}\psi_{\varepsilon}(x,t)=\partial_{i}\mathbf{u}_{\varepsilon}(x,t)\cdot\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}(x,t)\right).$
(2.39)

Moreover, we generalize the a.e. point-wise differential inequality (2.20) to

$|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|\leq\sqrt{2F(\mathbf{u}_{\varepsilon})}.$
(2.40)

To proceed, we define the phase-field analogues of the normal vector and the
mean curvature vector respectively by

$\displaystyle\mathbf{n}_{\varepsilon}(x,t)$
$\displaystyle:=\begin{cases}\frac{\nabla\psi_{\varepsilon}}{|\nabla\psi_{\varepsilon}|}(x,t)&\text{
if }\nabla\psi_{\varepsilon}(x,t)\neq 0,\\\ 0&\text{otherwise}.\end{cases}$
(2.41a) $\displaystyle\mathbf{H}_{\varepsilon}(x,t)$
$\displaystyle:=\begin{cases}-\left(\varepsilon\Delta\mathbf{u}_{\varepsilon}-\frac{1}{\varepsilon}\partial
F(\mathbf{u}_{\varepsilon})\right)\cdot\frac{\nabla\mathbf{u}_{\varepsilon}}{\left|\nabla\mathbf{u}_{\varepsilon}\right|}&\text{
if }\nabla\mathbf{u}_{\varepsilon}\neq 0,\\\ 0&\text{otherwise}.\end{cases}$
(2.41b)

Note that in (2.41b), the inner product is made with the column vectors of
$\nabla\mathbf{u}_{\varepsilon}=(\partial_{1}\mathbf{u}_{\varepsilon},\cdots,\partial_{d}\mathbf{u}_{\varepsilon})$.
We also define the orthogonal projection:

$\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon}:=\left\\{\begin{split}\big{(}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})\big{)}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})&~{}\text{
if }~{}\mathbf{u}_{\varepsilon}\in B_{\delta_{0}}(\mathfrak{m}),\\\
\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|}\right)\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|}&~{}\text{
if }~{}\mathbf{u}_{\varepsilon}\notin B_{\delta_{0}}\text{ and
}\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\neq 0,\\\ 0&~{}\text{ if
}~{}\mathbf{u}_{\varepsilon}\notin B_{\delta_{0}}\text{ and
}\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})=0\end{split}\right.$ (2.42)

where $\partial\mathrm{d}_{F}$ is interpreted as the generalized differential
(2.37) in case $\mathrm{d}_{F}$ is not classically differentiable at
$\mathbf{u}_{\varepsilon}$.

######  Lemma 2.8.

The following two identities holds:

$\displaystyle|\nabla\psi_{\varepsilon}|$
$\displaystyle=|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}||\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|$
$\displaystyle\text{ for any }(x,t),$ (2.43)
$\displaystyle\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}$
$\displaystyle=\frac{|\nabla\psi_{\varepsilon}|}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}}\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\otimes\mathbf{n}_{\varepsilon}$
$\displaystyle\text{ if }\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\neq
0,$ (2.44)

where in (2.44) the projection applies to each component of
$\nabla\mathbf{u}_{\varepsilon}$.

###### Proof.

Concerning (2.43), if $\mathbf{u}_{\varepsilon}\in
B_{\delta_{0}}(\mathfrak{m})$, then according to Lemma 2.7, $\mathrm{d}_{F}$
is $C^{1}$ and the generalized differential (2.37) coincide with the classical
one, and
$\frac{\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)}{|\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)|}|$
is well-defined. Thus

$\displaystyle\partial_{i}\psi_{\varepsilon}\overset{\eqref{ADM chain
rule}}{=}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\frac{\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)}{|\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)|}|\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)|\overset{\eqref{normalize
pdf}}{=}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})|\partial\mathrm{d}_{F}\left(\mathbf{u}_{\varepsilon}\right)|$
(2.45)

This together with
$|\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})|=1$ and the
first case in (2.42) leads to (2.43). If $\mathbf{u}_{\varepsilon}\notin
B_{\delta_{0}}$ and $\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\neq 0$,
then the second case in (2.42) leads to (2.43). If
$\mathbf{u}_{\varepsilon}\notin B_{\delta_{0}}$ and
$\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})=0$, then by (2.37), we have
$\nabla\psi_{\varepsilon}=0$ too. This finishes the proof of (2.43).

The statement (2.44) holds when $\nabla\psi_{\varepsilon}=0$ because (2.43)
then implies $\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}=0$.
When $\nabla\psi_{\varepsilon}\neq 0$, then

$\displaystyle\frac{|\nabla\psi_{\varepsilon}|}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}}\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\otimes\mathbf{n}_{\varepsilon}\overset{\eqref{normal
diff}}{=}\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}}\otimes\nabla\psi_{\varepsilon}$
$\displaystyle\overset{\eqref{ADM chain
rule}}{=}\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}}\otimes\left(\nabla\mathbf{u}_{\varepsilon}\cdot\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\right).$
(2.46)

When $\mathbf{u}_{\varepsilon}\notin B_{\delta_{0}}$, this last term is
exactly the second case defining
$\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}$. When
$\mathbf{u}_{\varepsilon}\in B_{\delta_{0}}$, by (2.19) it simplifies to

$\big{(}\nabla\mathbf{u}_{\varepsilon}\cdot\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})\big{)}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{\varepsilon})\overset{\eqref{projection1}}{=}\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}.$
(2.47)

This finish the proof of (2.44). ∎

As we shall not integrate the time variable $t$ throughout this section, we
shall abbreviate the spatial integration $\int_{\Omega}$ by $\int$ and
sometimes we omit the $\,dx$. The following lemma gives various coercivity
estimates of $E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]$ (1.11). It was
due to [21], generalizing the one by [16] to vectorial cases. We present the
proof for the convenience of the readers.

######  Lemma 2.9.

There exists a universal constant $C>0$ which is independent of $t\in[0,T)$
and $\varepsilon$ such that the following estimates hold for every
$t\in(0,T)$:

$\displaystyle\int\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\right)\,dx$
$\displaystyle\leq E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma],$ (2.48a)
$\displaystyle\varepsilon\int\left(\left|\nabla\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}\right)\,dx$
$\displaystyle\leq 2E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma],$ (2.48b)
$\displaystyle\int\left(\sqrt{\varepsilon}\left|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|-\frac{1}{\sqrt{\varepsilon}}\left|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\right|\right)^{2}\,dx$
$\displaystyle\leq 2E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma],$ (2.48c)
$\displaystyle\int\left({\frac{\varepsilon}{2}}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})+\left|\nabla\psi_{\varepsilon}\right|\right)\left(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\right)\,dx$
$\displaystyle\leq CE_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma],$ (2.48d)
$\displaystyle\int\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})+|\nabla\psi_{\varepsilon}|\right)\min\left(\mathrm{d}_{\Sigma}^{2},1\right)\,dx$
$\displaystyle\leq CE_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma].$ (2.48e)

###### Proof.

Using (2.41a), we obtain
$\nabla\psi_{\varepsilon}=|\nabla\psi_{\varepsilon}|\mathbf{n}_{\varepsilon}$.
Note also that (2.42) implies

$\left|\nabla\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\left|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}=\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}.$
(2.49)

Altogether, we can write

$\displaystyle E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]=$
$\displaystyle\int\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|+\int|\nabla\psi_{\varepsilon}|(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon})$
$\displaystyle=$
$\displaystyle\frac{\varepsilon}{2}\int\left|\nabla\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}$
$\displaystyle+\int\frac{\varepsilon}{2}\left|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|$
$\displaystyle+\int|\nabla\psi_{\varepsilon}|(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}).$
(2.50)

By (2.40) and (2.43), the second last integral has non-negative integrand. and
the claim follows from and completing a square. This also yields (2.48a),
(2.48b) and (2.48c).

Combining (2.48a) with
$E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]\geq\int\left(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\right)\left|\nabla\psi_{\varepsilon}\right|$
and $1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\leq 2$ yields (2.48d). Finally,
by (2.29) and $\delta_{0}\in(0,1)$ we have

$1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\geq
1-\phi\left(\frac{\mathrm{d}_{\Sigma}}{\delta_{0}}\right)\geq\min\left(\frac{\mathrm{d}_{\Sigma}^{2}}{2\delta_{0}^{2}},1-\phi(\tfrac{1}{2})\right)\geq
C_{\phi,\delta_{0}}\min(\mathrm{d}_{\Sigma}^{2},1).$ (2.51)

This together with (2.48d) implies (2.48e). ∎

The following result was first proved in [16] for the scalar case, and was
generalized to a matrix-valued one in [21]. We present the proof in Appendix A
for the convenience of the readers

######  Proposition 2.10.

There exists a constant $C=C(\Sigma_{t})$ depending on the interface
$\Sigma_{t}$ such that

$\displaystyle\frac{d}{dt}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]$
$\displaystyle+\frac{1}{2\varepsilon}\int\left(\varepsilon^{2}|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}-|\mathbf{H}_{\varepsilon}|^{2}\right)\,dx+\frac{1}{2\varepsilon}\int\Big{|}\mathbf{H}_{\varepsilon}-\varepsilon|\nabla\mathbf{u}_{\varepsilon}|\mathbf{H}\Big{|}^{2}\,dx$
$\displaystyle+\frac{1}{2\varepsilon}\int\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\Big{|}^{2}\,dx\leq
CE_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma].$ (2.52)

##  3\. Estimates of level sets

The main task of this section is to derive the convergence rate estimate
(1.17) and use it to obtain fine estimates of the level sets of
$\psi_{\varepsilon}$. We start with a corollary of Proposition 2.10.

######  Lemma 3.1.

There exists a universal constant $C=C(\Sigma_{0})$ such that

$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}{F(\mathbf{u}_{\varepsilon})}-\bm{\xi}\cdot\nabla\psi_{\varepsilon}\right)\,dx\leq
C\varepsilon,$ (3.1a)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}\left(\left|\nabla\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}\right|^{2}\right)\,dx+\int_{0}^{T}\int_{\Omega}\left(\left|\partial_{t}\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\partial_{t}\mathbf{u}_{\varepsilon}\right|^{2}\right)\,dxdt\leq
C,$ (3.1b)
$\displaystyle\sup_{t\in[0,T]}A_{\varepsilon}(\mathbf{u}_{\varepsilon}(\cdot,t))+\sup_{t\in[0,T]}\|\nabla\psi_{\varepsilon}(\cdot,t)\|_{L^{1}(\Omega)}\leq
C,$ (3.1c)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega}|\nabla\psi_{\varepsilon}|-\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx\leq
C\varepsilon,.$ (3.1d)

Moreover, for any fixed $\delta\in(0,\delta_{I})$, there holds

$\displaystyle\sup_{t\in[0,T]}\int_{\Omega^{\pm}_{t}\backslash
B_{\delta}(\Sigma_{t})}\left(|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{{F(\mathbf{u}_{\varepsilon})}}{\varepsilon^{2}}\right)\,dx\leq\delta^{-2}C,$
(3.2a) $\displaystyle\int_{0}^{T}\int_{\Omega^{\pm}_{t}\backslash
B_{\delta}(\Sigma_{t})}|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dxdt\leq\delta^{-2}C.$
(3.2b)

###### Proof.

To prove (3.1b), it follows from (2.52) and the assumption (1.15) that

$\displaystyle\sup_{t\in[0,T]}\frac{1}{\varepsilon}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](t)+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\int_{\Omega}\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})(\nabla\cdot\bm{\xi})\Big{|}^{2}\,dxdt$
$\displaystyle+\frac{1}{\varepsilon^{2}}\int_{0}^{T}\int_{\Omega}\left(\varepsilon^{2}|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}-|\mathbf{H}_{\varepsilon}|^{2}+\Big{|}\mathbf{H}_{\varepsilon}-\varepsilon\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\Big{|}^{2}\right)\,dxdt$
$\displaystyle\qquad\leq\frac{1}{\varepsilon}e^{(1+T)C(\Sigma_{0})}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma](0)\leq
e^{(1+T)C(\Sigma_{0})}.$ (3.3)

Now we show the third term on the left hand side of (3) has non-negative
integrand. By (1.7a) and (2.41b) we have
$\mathbf{H}_{\varepsilon}=-\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}\cdot\frac{\nabla\mathbf{u}_{\varepsilon}}{|\nabla\mathbf{u}_{\varepsilon}|}$.
Using this, we can expand the second integrand in (3) and apply the Cauchy-
Schwarz inequality to obtain

$\displaystyle~{}\varepsilon^{2}|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}-|\mathbf{H}_{\varepsilon}|^{2}+\Big{|}\mathbf{H}_{\varepsilon}-\varepsilon\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\Big{|}^{2}$
$\displaystyle=$
$\displaystyle~{}\varepsilon^{2}|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}+\varepsilon^{2}|\mathbf{H}|^{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+2\varepsilon^{2}(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}\cdot\partial_{t}\mathbf{u}_{\varepsilon}$
$\displaystyle\geq$
$\displaystyle~{}\varepsilon^{2}|\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}|^{2}.$

This leads to (3.1a), and together with (3) implies

$\int_{0}^{T}\int_{\Omega}|\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}|^{2}\,dxdt\leq
e^{(1+T)C(\Sigma_{0})}.$ (3.4)

On the other hand, using the orthogonal projection (2.42), we obtain

$\displaystyle\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})(\nabla\cdot\bm{\xi})\Big{|}^{2}=\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-\varepsilon\Pi_{\mathbf{u}_{\varepsilon}}\partial_{t}\mathbf{u}_{\varepsilon}\Big{|}^{2}+\Big{|}\varepsilon\Pi_{\mathbf{u}_{\varepsilon}}\partial_{t}\mathbf{u}_{\varepsilon}-\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})(\nabla\cdot\bm{\xi})\Big{|}^{2}.$

This together with (3) yields

$\displaystyle\frac{1}{\varepsilon^{2}}\int_{0}^{T}\int_{\Omega}\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-\varepsilon\Pi_{\mathbf{u}_{\varepsilon}}\partial_{t}\mathbf{u}_{\varepsilon}\Big{|}^{2}\leq
e^{(1+T)C(\Sigma_{0})}.$ (3.5)

The above two estimates together with (2.48c) implies (3.1b). Concerning
(3.1c),

$A_{\varepsilon}(\mathbf{u}_{\varepsilon}(\cdot,0))\overset{\eqref{dissipation}}{=}A_{\varepsilon}(\mathbf{u}_{\varepsilon})\overset{\eqref{projection1},\eqref{eq:2.7global}}{\geq}\int_{\Omega}\left(\frac{\varepsilon}{2}|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{2\varepsilon}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}\right)\,dx\geq\int_{\Omega}|\nabla\psi_{\varepsilon}|\,dx.$
(3.6)

The estimate (3.1d) follows from (3.1b) and (2.48d), and (3.1a) follows from
(2.48b). Finally, combining (3.2a) with (3.4) leads us to (3.2b). ∎

We shall use (3.2) together with the method of Chen–Struwe [8] to show that
the weak limit of $\mathbf{u}_{\varepsilon}$ are harmonic heat flows from the
bulk regions $\Omega_{t}^{\pm}$ to $\mathfrak{m}_{\pm}$ respectively. However,
the bulk potential (1.5) depends on the relative distances to these two
manifolds, and we must find a quantitative way to distinguish them. This is
done in the following:

######  Theorem 3.2.

Under the assumptions of Theorem 1.1, there exists $C>$ independent of
$\varepsilon$ so that

$\int_{\Omega}|\psi_{\varepsilon}-c_{F}\mathbf{1}_{\Omega_{t}^{+}}|\,dx\leq
C\varepsilon^{1/2}$ (3.7)

###### Proof.

The proof will be done in two steps.

Step 1: derivation of differential inequalities. Let
$\chi(x,t)=\mathbf{1}_{\Omega_{t}^{+}}-\mathbf{1}_{\Omega_{t}^{-}}$ and let
$\eta(\cdot)$ be the truncation of the identity map

$\eta(x)=\left\\{\begin{array}[]{rl}x\qquad\text{when
}&x\in[-\delta_{0},\delta_{0}],\\\ \delta_{0}\qquad\text{when
}&x\geq\delta_{0},\\\ -\delta_{0}\qquad\text{when
}&x\leq-\delta_{0},\end{array}\right.$ (3.8)

and we denote $\zeta=|\eta|$. It follows from (2.42) and the generalized chain
rule (2.39) that

$\displaystyle\partial_{t}\psi_{\varepsilon}=$
$\displaystyle\left(\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}\right)\cdot\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})-\mathbf{H}\cdot\nabla\psi_{\varepsilon}$
(3.9)

In the sequel, $(h)^{\pm}$ indicates the positive/negative part of a function
$h(x)$. By [10, pp. 153],

$\partial_{x_{i}}(h(x))^{+}=(\partial_{x_{i}}h(x))\mathbf{1}_{\\{x:h(x)>0\\}}(x)\quad\text{for
}a.e.~{}~{}x.$ (3.10)

So we can write, using the formula $f=f^{+}-f^{-}$, that

$2\psi_{\varepsilon}-c_{F}=2(\psi_{\varepsilon}-c_{F})^{+}+c_{F}-2(\psi_{\varepsilon}-c_{F})^{-}.$
(3.11)

We shall establish a differential inequality of the two energies which sum up
to (1.14):

$\displaystyle g_{\varepsilon}(t):=$
$\displaystyle\int\left(\psi_{\varepsilon}-c_{F}\right)^{+}\zeta\circ\mathrm{d}_{\Sigma}\,dx$
(3.12a) $\displaystyle h_{\varepsilon}(t):=$
$\displaystyle\int\Big{(}c_{F}\chi-
c_{F}+2(\psi_{\varepsilon}-c_{F})^{-}\Big{)}\eta\circ\mathrm{d}_{\Sigma}\,dx$
(3.12b)

Since $\psi_{\varepsilon}\geq 0$, we have
$(\psi_{\varepsilon}-c_{F})^{-}\in[0,c_{F}]$ and thus
$c_{F}-2(\psi_{\varepsilon}-c_{F})^{-}$ ranges in $[-c_{F},c_{F}]$. In view of
(3.8), we have $\eta\chi\geq 0$, so the integrands of these two energies are
both non-negative. By (1.15),

$g_{\varepsilon}(0)+h_{\varepsilon}(0)\lesssim\varepsilon.$ (3.13)

Now we proceed in the derivation of Gönwall’s inequalities of
$g_{\varepsilon}$ and $h_{\varepsilon}$. Using (3.9)

$\displaystyle g_{\varepsilon}^{\prime}(t)\overset{\eqref{volume evo1}}{=}$
$\displaystyle\int_{\\{\psi_{\varepsilon}>c_{F}\\}}(\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon})\cdot\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle-\int_{\\{\psi_{\varepsilon}>c_{F}\\}}\mathbf{H}\cdot\nabla\psi_{\varepsilon}\zeta(\mathrm{d}_{\Sigma})+\int\left(\psi_{\varepsilon}-c_{F}\right)^{+}\partial_{t}\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle\overset{\eqref{der positive part}}{=}$
$\displaystyle\int_{\\{\psi_{\varepsilon}>c_{F}\\}}(\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon})\cdot\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle-\int\mathbf{H}\cdot\nabla\left(\psi_{\varepsilon}-c_{F}\right)^{+}\zeta(\mathrm{d}_{\Sigma})-\int\left(\psi_{\varepsilon}-c_{F}\right)^{+}\mathbf{H}\cdot\nabla\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle+\int\left(\partial_{t}\zeta(\mathrm{d}_{\Sigma})+\mathbf{H}\cdot\nabla\zeta(\mathrm{d}_{\Sigma})\right)\left(\psi_{\varepsilon}-c_{F}\right)^{+}$

Finally by an integration by part, we can combine the second and the third
integral and obtain

$\displaystyle
g_{\varepsilon}^{\prime}(t)\overset{\eqref{eq:2.7global}}{\leq}$
$\displaystyle\int_{\\{\psi_{\varepsilon}>c_{F}\\}}\left|(\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon})\cdot\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|}\sqrt{2F(\mathbf{u}_{\varepsilon})}\right|\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle+\int(\operatorname{div}\mathbf{H})\left(\psi_{\varepsilon}-c_{F}\right)^{+}\zeta(\mathrm{d}_{\Sigma})+\int\left(\partial_{t}\zeta(\mathrm{d}_{\Sigma})+\mathbf{H}\cdot\nabla\zeta(\mathrm{d}_{\Sigma})\right)\left(\psi_{\varepsilon}-c_{F}\right)^{+}$
$\displaystyle\overset{\eqref{mcf}}{\leq}\int\varepsilon\Big{|}\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}\Big{|}^{2}+\int\frac{1}{\varepsilon}{F(\mathbf{u}_{\varepsilon})}\zeta^{2}(\mathrm{d}_{\Sigma})+Cg_{\varepsilon}(t)$
$\displaystyle\overset{\eqref{energy
bound3}}{\leq}\int\varepsilon\Big{|}\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}\Big{|}^{2}+CE_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]+Cg_{\varepsilon}(t).$

In view of (3.4), we can apply Grönwall’s lemma and obtain
$g_{\varepsilon}(t)\leq C\varepsilon$.

Similar calculation shows $h_{\varepsilon}^{\prime}(t)\leq
Ch_{\varepsilon}(t)$. For simplicity we denote

$c_{F}\chi-c_{F}+2(\psi_{\varepsilon}-c_{F})^{-}=:w_{\varepsilon}.$

Using $\partial_{i}\chi\eta(\mathrm{d}_{\Sigma})\equiv 0$ (in the sense of
distributions), we find

$\partial_{i}w_{\varepsilon}\eta(\mathrm{d}_{\Sigma})=2\partial_{i}\psi_{\varepsilon}\mathbf{1}_{\\{\psi_{\varepsilon}<c_{F}\\}}\eta(\mathrm{d}_{\Sigma})\quad\text{for
}a.e.~{}~{}x.$ (3.14)

So by the same calculation for $g_{\varepsilon}$ we obtain

$\displaystyle h_{\varepsilon}^{\prime}(t)\leq$
$\displaystyle\int_{\\{\psi_{\varepsilon}<c_{F}\\}}2\left|(\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon})\cdot\frac{\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})}{|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|}\sqrt{2F(\mathbf{u}_{\varepsilon})}\right|\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle+\int(\operatorname{div}\mathbf{H})w_{\varepsilon}\eta(\mathrm{d}_{\Sigma})+\int\Big{(}\partial_{t}\eta(\mathrm{d}_{\Sigma})+\mathbf{H}\cdot\nabla\eta(\mathrm{d}_{\Sigma})\Big{)}w_{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\int\varepsilon\Big{|}\partial_{t}\mathbf{u}_{\varepsilon}+(\mathbf{H}\cdot\nabla)\mathbf{u}_{\varepsilon}\Big{|}^{2}+CE_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]+Ch_{\varepsilon}(t),$

and $h_{\varepsilon}(t)\leq C\varepsilon$ follows from the Grönwall lemma.
Finally,

$\displaystyle\int|2\psi_{\varepsilon}-c_{F}-c_{F}\chi|\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle\overset{\eqref{psi deco}}{\leq}\int
2(\psi_{\varepsilon}-c_{F})^{+}\zeta(\mathrm{d}_{\Sigma})+\int\Big{|}c_{F}-2(\psi_{\varepsilon}-c_{F})^{-}-c_{F}\chi\Big{|}\zeta(\mathrm{d}_{\Sigma})$
$\displaystyle=2g_{\varepsilon}+h_{\varepsilon}\leq C\varepsilon.$ (3.15)

Step 2: pass to the unweighted inequality. We first note that (3.15) implies
$\eqref{volume convergence}$ with $\Omega$ being replaced by $\Omega\backslash
B_{\delta_{0}}(\Sigma_{t})$. So we shall focus on the estimate in
$B_{\delta_{0}}(\Sigma_{t})$. We shall use the following elementary estimate

$\left(\int_{0}^{\delta_{0}}|f(r)|\,dr\right)^{2}\leq
2\|f\|_{\infty}\int_{0}^{\delta_{0}}|f(r)|r\,dr\quad\forall f\in
L^{\infty}\left(0,\delta_{0}\right).$ (3.16)

Let $\chi_{\varepsilon}=2\psi_{\varepsilon}-c_{F}$. For each fixed
$p\in\Sigma_{t}$, we can apply the above inequality and yield

$\displaystyle\left(\int_{0}^{\delta_{0}}\left|\chi(p+y\mathbf{n},t)-\chi_{\varepsilon}(p+y\mathbf{n},t)\right|\,dy\right)^{2}$
$\displaystyle=$
$\displaystyle\int_{0}^{\delta_{0}}\left|\chi(p+y\mathbf{n},t)-\chi_{\varepsilon}(p+y\mathbf{n},t)\right|\mathrm{d}_{\Sigma}\left(p+y\mathbf{n},t\right)\,dy$

This allows us to estimate for the $\delta_{0}$ -neighborhood of $\Sigma_{t}$

$\displaystyle\left(\int_{B_{\delta_{0}}(\Sigma_{t})}\left|\chi(x,t)-\chi_{\varepsilon}(x,t)\right|\,dx\right)^{2}$
$\displaystyle\lesssim$
$\displaystyle\left(\sum_{\pm}\int_{\Sigma_{t}}\int_{0}^{\delta_{0}}\left|\chi\left(p\pm
y\mathbf{n},t\right)-\chi_{\varepsilon}(p\pm
y\mathbf{n},t)\right|\,dydS(p)\right)^{2}$ $\displaystyle\lesssim$
$\displaystyle\int_{\Sigma_{t}}\int_{-\delta_{0}}^{\delta_{0}}\left|\chi\left(p+y\mathbf{n},t\right)-\chi_{\varepsilon}\left(p+y\mathbf{n},t\right)\right|\mathrm{d}_{\Sigma}\left(p+y\mathbf{n},t\right)\,dydS(p)$
$\displaystyle\lesssim$
$\displaystyle\int_{B_{\delta_{0}}(\Sigma_{t})}\left|\chi(x,t)-\chi_{\varepsilon}(x,t)\right|\mathrm{d}_{\Sigma}(x,t)\,dx$

This gives the proof in $B_{\delta_{0}}(\Sigma_{t})$. ∎

######  Corollary 3.3.

There exists a sequence of $\varepsilon_{k}\downarrow 0$ and
$\mathbf{u}^{\pm}(x,t)$ with

$\displaystyle\mathbf{u}^{\pm}\in L^{\infty}(0,T;L^{\infty}(\Omega)\cap
H^{1}_{loc}(\Omega^{+}_{t};\mathfrak{m}_{\pm})),\partial_{t}\mathbf{u}^{\pm}\in
L^{2}(0,T;L^{2}_{loc}(\Omega^{+}_{t};\mathfrak{m}_{\pm}))$ (3.17)

such that $\mathbf{u}_{k}:=\mathbf{u}_{\varepsilon_{k}}$ satisfies

$\displaystyle\partial_{t}\mathbf{u}_{k}\xrightarrow{k\to\infty}\partial_{t}\mathbf{u}^{\pm}$
$\displaystyle~{}\text{weakly in}~{}L^{2}(0,T;L^{2}_{loc}(\Omega^{\pm}_{t})),$
(3.18a)
$\displaystyle\nabla\mathbf{u}_{k}\xrightarrow{k\to\infty}\nabla\mathbf{u}^{\pm}$
$\displaystyle~{}\text{weakly
in}~{}L^{\infty}(0,T;L^{2}_{loc}(\Omega^{\pm}_{t})),$ (3.18b)
$\displaystyle\mathbf{u}_{k}\xrightarrow{k\to\infty}\mathbf{u}^{\pm}$
$\displaystyle~{}\text{strongly in}~{}C([0,T];L^{2}_{loc}(\Omega^{\pm}_{t})).$
(3.18c)

###### Proof.

It follows from (1.22), (3.2a) and (3.2b) that, for any
$\delta\in(0,\delta_{0})$, there exists a subsequence
$\varepsilon_{k}=\varepsilon_{k}(\delta)>0$ such that

$\displaystyle\mathbf{u}_{\varepsilon_{k}}\xrightarrow{k\to\infty}\mathbf{u}^{\pm}$
$\displaystyle~{}\text{weakly-star in}~{}L^{\infty}(0,T;L^{\infty}(\Omega)),$
(3.19a)
$\displaystyle\partial_{t}\mathbf{u}_{\varepsilon_{k}}\xrightarrow{k\to\infty}\partial_{t}\bar{\mathbf{u}}_{\delta}^{\pm}$
$\displaystyle~{}\text{weakly in}~{}L^{2}(0,T;L^{2}(\Omega^{\pm}_{t}\backslash
B_{\delta}(\Sigma_{t}))),$ (3.19b)
$\displaystyle\nabla\mathbf{u}_{\varepsilon_{k}}\xrightarrow{k\to\infty}\nabla\bar{\mathbf{u}}_{\delta}^{\pm}$
$\displaystyle~{}\text{weakly-star
in}~{}L^{\infty}(0,T;L^{2}(\Omega^{\pm}_{t}\backslash
B_{\delta}(\Sigma_{t}))),$ (3.19c)

and $\mathbf{u}^{\pm}=\bar{\mathbf{u}}_{\delta}^{\pm}$ a.e. in
$U_{\pm}(\delta):=\cup_{t\in[0,T]}\left(\Omega_{t}^{\pm}\backslash
B_{\delta}(\Sigma_{t})\right)\times\\{t\\}$. This combined with (3.19b) and
(3.19c) leads to

$\mathbf{u}\in
L^{\infty}(0,T;H^{1}_{loc}(\Omega^{\pm}_{t}))~{}\text{with}~{}\partial_{t}\mathbf{u}\in
L^{2}(0,T;L^{2}_{loc}(\Omega^{\pm}_{t})).$ (3.20)

It remains to show that $\mathbf{u}^{\pm}$ are mappings into
$\mathfrak{m}_{\pm}$. By a diagonal argument we obtain (3.18). Using (3.18c),
(3.2a), and Fatou’s lemma, we deduce that $F(\mathbf{u})=0$ a.e. in $\Omega$.
In view of (1.4) we deduce that the images of $\mathbf{u}^{\pm}$ lies in
$\mathfrak{m}$. By (3.18c) and (1.22)

$\psi_{\varepsilon_{k}}\overset{\eqref{psi}}{=}\mathrm{d}_{F}\circ\mathbf{u}_{k}\xrightarrow{k\to\infty}\mathrm{d}_{F}\circ\mathbf{u}^{\pm}\text{
strongly in }C([0,T];L^{2}_{loc}(\Omega^{\pm}_{t})).$ (3.21)

This together with (3.7) and (2.23) yields that $\mathbf{u}^{\pm}$ maps into
$\mathfrak{m}_{\pm}$ respectively. Combining this with (3.20) yields (3.17). ∎

######  Lemma 3.4.

For any $\delta\in(0,\delta_{0})$, there exist three numbers
$b^{\pm}_{\delta}\in[\delta,2\delta]$ s.t. the sets

$\\{x:\psi_{\varepsilon}>c_{F}-b^{+}_{\delta}\\}\text{ and
}\\{x:\psi_{\varepsilon}<b^{-}_{\delta}\\}$ (3.22)

have finite perimeters and

$\displaystyle\left|\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=c_{F}-b^{+}_{\delta}\\}\right)-\mathcal{H}^{d-1}(\Sigma_{t})\right|$
$\displaystyle+\left|\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=b^{-}_{\delta}\\}\right)-\mathcal{H}^{d-1}(\Sigma_{t})\right|\leq
C\varepsilon^{1/2}\delta^{-1}.$ (3.23)

###### Proof.

For any $\delta<\delta_{0}\ll c_{F}$, we denote 111 Note that this definition
only applies within the proof of the lemma.

$\Omega_{t}^{\varepsilon,\delta}=\\{x\in\Omega:c_{F}-2\delta<\psi_{\varepsilon}(x,t)<c_{F}-\delta\\}$
(3.24)

Recall the co-area formula of BV function [10, section 5.5] which states that
the (measure-theoretic) boundary $\partial\Omega_{t}^{\varepsilon,\delta,\pm}$
has finite perimeter for almost every $\delta$. Note that we shall denote the
corresponding (measure-theoretic) outer normal vector by $\nu$. Using (3.1b)
and (2.48d), we deduce for almost every $\delta\in(0,\delta_{0})$ that

$\displaystyle C\varepsilon\overset{\eqref{calibration est1}}{\geq}$
$\displaystyle\int_{\Omega_{t}^{\varepsilon,\delta}}\left(|\nabla\psi_{\varepsilon}|-\bm{\xi}\cdot\nabla\psi_{\varepsilon}\right)\,dx\qquad(\geq
0)$ $\displaystyle=$
$\displaystyle\int_{c_{F}-2\delta}^{c_{F}-\delta}\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=s\\}\right)\,ds-\int_{\partial\Omega_{t}^{\varepsilon,\delta}}\bm{\xi}\cdot\nu\psi_{\varepsilon}\,d\mathcal{H}^{d-1}+\int_{\Omega_{t}^{\varepsilon,\delta}}(\operatorname{div}\bm{\xi})\psi_{\varepsilon}\,dx$

where $\nu$ is the outward unit normal of the set under integration, defined
on its (measure-theoretic) boundaries. So we obtain

$\displaystyle\Big{|}\int_{c_{F}-2\delta}^{c_{F}-\delta}\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=s\\}\right)\,ds-\int_{\partial\Omega_{t}^{\varepsilon,\delta}}\bm{\xi}\cdot\nu\psi_{\varepsilon}\,d\mathcal{H}^{d-1}\Big{|}\lesssim\varepsilon+|\Omega_{t}^{\varepsilon,\delta}|$
(3.25)

Applying the divergence theorem and adding zero, the second integral on the
LHS writes

$\displaystyle\int_{\partial\Omega_{t}^{\varepsilon,\delta}}\bm{\xi}\cdot\nu\psi_{\varepsilon}\,d\mathcal{H}^{d-1}$
$\displaystyle=$
$\displaystyle\left(c_{F}-2\delta\right)\int_{\\{\psi_{\varepsilon}>c_{F}-2\delta\\}}\operatorname{div}\bm{\xi}\,dx+(c_{F}-\delta)\int_{\\{\psi_{\varepsilon}<c_{F}-\delta\\}}\operatorname{div}\bm{\xi}\,dx$
$\displaystyle\underbrace{-\left(c_{F}-2\delta\right)\int_{\Omega_{t}^{+}}\operatorname{div}\bm{\xi}\,dx-(c_{F}-\delta)\int_{\Omega_{t}^{-}}\operatorname{div}\bm{\xi}\,dx+\delta\mathcal{H}^{d-1}(\Sigma_{t})}_{=0}$
(3.26)

Note that by the orientation of $\nu$ and $\bm{\xi}$, we see the three terms
in the last line of (3) sum up to zero. So merging terms having same
prefactors on the RHS of (3), and then substituting (3) into (3.25) yields

$\displaystyle\left|\int_{c_{F}-2\delta}^{c_{F}-\delta}\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=s\\}\right)\,ds-\delta\mathcal{H}^{d-1}(\Sigma_{t})\right|$
$\displaystyle\lesssim\varepsilon+|\Omega_{t}^{\varepsilon,\delta}+\Big{|}\Omega_{t}^{+}\triangle{\\{x:\psi_{\varepsilon}>c_{F}-2\delta\\}}\Big{|}+\Big{|}\Omega_{t}^{-}\triangle{\\{x:\psi_{\varepsilon}<c_{F}-\delta\\}}\Big{|}$
(3.27)

where $A\triangle B=(A-B)\cup(B-A)$ is the symmetric difference of two sets
$A,B$. Up to a constant, we rewrite the last two terms by

$\displaystyle r_{\varepsilon}^{+}:=$
$\displaystyle~{}\Big{|}\Omega_{t}^{+}\triangle{\\{x:\psi_{\varepsilon}>c_{F}-2\delta\\}}\Big{|}$
$\displaystyle=$
$\displaystyle~{}\Big{|}{\\{x\in\Omega_{t}^{+}:\psi_{\varepsilon}\leq
c_{F}-2\delta\\}}\Big{|}+\Big{|}{\\{x\in\Omega_{t}^{-}:\psi_{\varepsilon}>c_{F}-2\delta\\}},$
$\displaystyle r_{\varepsilon}^{-}:=$
$\displaystyle~{}\Big{|}\Omega_{t}^{-}\triangle{\\{x:\psi_{\varepsilon}<c_{F}-\delta\\}}\Big{|}$
$\displaystyle=$
$\displaystyle~{}\Big{|}{\\{x\in\Omega_{t}^{-}:\psi_{\varepsilon}\geq
c_{F}-\delta\\}}\Big{|}+\Big{|}{\\{x\in\Omega_{t}^{+}:\psi_{\varepsilon}<c_{F}-\delta\\}}\Big{|}.$

Now using the Chebyshev inequality and (3.7) yields
$r_{\varepsilon}^{-}+r_{\varepsilon}^{+}\lesssim\varepsilon^{1/2}\delta^{-1}$,
and substitute this estimate into (3) leads

$\displaystyle\left|\frac{1}{\delta}\int_{c_{F}-2\delta}^{c_{F}-\delta}\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=s\\}\right)\,ds-\mathcal{H}^{d-1}(\Sigma_{t})\right|\leq
C\varepsilon^{1/2}\delta^{-1}.$ (3.28)

So the existence of $b^{+}_{\delta}\in[\delta,2\delta]$ satisfying (3.23)
follows by Fubini’s theorem. The other case can be proved in the same way and
we omit the proofs here. ∎

######  Proposition 3.5.

There exists a subsequence $\varepsilon_{k}\downarrow 0$ and
$b^{\pm}_{k}\in[1/k,2/k]$ so that the sets

$\displaystyle\Omega_{t}^{k,+}$
$\displaystyle=\\{x\in\Omega:\psi_{k}(x,t)>c_{F}-b^{+}_{k}\\},$ (3.29a)
$\displaystyle\Omega_{t}^{k,-}$
$\displaystyle=\\{x\in\Omega:\psi_{k}(x,t)<b^{-}_{k}\\}$ (3.29b)

have uniformly bounded perimeter where $\psi_{k}=\psi_{\varepsilon_{k}}$, and

$\displaystyle\left|\mathcal{H}^{d-1}(\partial\Omega_{t}^{k,\pm})-\mathcal{H}^{d-1}(\Sigma_{t})\right|\leq
C\varepsilon_{k}^{1/4},$ (3.30)
$\displaystyle\mathbf{1}_{\Omega_{t}^{k,\pm}}\xrightarrow{k\to\infty}\mathbf{1}_{\Omega_{t}^{\pm}}\text{
weakly-star in }BV(\Omega).$ (3.31)

Moreover there exists some $K_{1}>0$ so that for any $k>K_{1}$, the solution
$\mathbf{u}_{k}:=\mathbf{u}_{\varepsilon_{k}}$ satisfies

$\displaystyle\mathbf{u}_{k}(\Omega_{t}^{k,\pm})\subset
B_{\delta_{0}}(\mathfrak{m}_{\pm}),$ (3.32)
$\displaystyle\sup_{t\in[0,T]}\int_{\Omega_{t}^{k,\pm}}\Big{|}\nabla
P_{\mathfrak{m}}(\mathbf{u}_{k})\Big{|}^{2}\,dx+\int_{0}^{T}\int_{\Omega_{t}^{k,\pm}}\Big{|}\partial_{t}P_{\mathfrak{m}}(\mathbf{u}_{k})\Big{|}^{2}\,dxdt\leq
e^{(1+T)C(\Sigma_{0})}$ (3.33)

###### Proof.

For each $k\in\mathbb{N},$ larger than the integer part of $2/\delta_{0}$, we
choose $\delta=1/k$ in Lemma 3.4. This yields $b_{k}^{\pm}\in[1/k,2/k]$ so
that

$\left|\mathcal{H}^{d-1}\left(\\{x:\psi_{\varepsilon}=c_{F}-b^{+}_{k}\\}\right)-\mathcal{H}^{d-1}(\Sigma_{t})\right|\leq\varepsilon^{1/2}k$
(3.34)

Choosing $\varepsilon_{k}\in(0,k^{-4})$ leads to the ‘plus’ cases of (3.30).
By (3.7) we have for each fixed $k$ that

$\mathbf{1}_{\\{x:\psi_{\varepsilon}>c_{F}-b^{+}_{k}\\}}\xrightarrow{\varepsilon\to
0}\mathbf{1}_{\Omega_{t}^{+}}\text{ strongly in }L^{1}(\Omega).$ (3.35)

So by a diagonal argument, we find $\varepsilon_{k}\in(0,k^{-4})$ so that

$\mathbf{1}_{\Omega_{t}^{k,+}}\xrightarrow{k\to\infty}\mathbf{1}_{\Omega_{t}^{+}}\text{
strongly in }L^{1}(\Omega).$ (3.36)

This combined with (3.30) implies the ‘plus case of (3.31). The ‘minus’ cases
can be done in the same way.

Concerning (3.33), by (2.18) and $b_{k}^{\pm}\to 0$, there exists $K_{1}>0$ so
that for any $k\geq K_{1}$ there holds

$\displaystyle\mathrm{d}_{F}\circ\mathbf{u}_{k}>c_{F}-b_{k}^{+}\text{ implies
}\mathbf{u}_{k}\in B_{\delta_{0}}(\mathfrak{m}_{+}),$
$\displaystyle\mathrm{d}_{F}\circ\mathbf{u}_{k}<b_{k}^{-}\text{ implies
}\mathbf{u}_{k}\in B_{\delta_{0}}(\mathfrak{m}_{-}).$

So for $k\geq K_{1}$, we have (3.32) and the nearest point projection to
$\mathfrak{m}$ is smooth. So we have

$\mathbf{u}_{k}=P_{\mathfrak{m}}(\mathbf{u}_{k})+\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k}),$
(3.37)

where $\mathrm{d}_{\mathfrak{m}}$ is the signed-distance function, which is
defined by (2.17) and is smooth in $B_{\delta_{0}}(\mathfrak{m})$.
Differentiating the above equation gives

$\partial_{x_{i}}\mathbf{u}_{k}=\partial_{x_{i}}\left(P_{\mathfrak{m}}(\mathbf{u}_{k})\right)+\partial_{x_{i}}\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})+\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})\partial_{x_{i}}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k}).$
(3.38)

Note that the last term is orthogonal to
$\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})$ by
$|\partial\mathrm{d}_{\mathfrak{m}}|=1$. So we have

$\displaystyle|\partial_{x_{i}}\mathbf{u}_{k}|^{2}$
$\displaystyle=|\partial_{x_{i}}P_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}+|\partial_{x_{i}}\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}+\mathrm{d}_{\mathfrak{m}}^{2}(\mathbf{u}_{k})|\partial_{x_{i}}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}$
$\displaystyle\quad+2\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})\partial_{x_{i}}\left(P_{\mathfrak{m}}(\mathbf{u}_{k})\right)\cdot\partial_{x_{i}}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})$
$\displaystyle\geq(1-C\delta_{0})|\partial_{x_{i}}P_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}+|\partial_{x_{i}}\mathbf{u}_{k}\cdot\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}+\mathrm{d}_{\mathfrak{m}}^{2}(\mathbf{u}_{k})|\partial_{x_{i}}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}.$
(3.39)

On the other hand, by (2.42)

$|\Pi_{\mathbf{u}}\partial_{x_{i}}\mathbf{u}_{k}|^{2}=|\partial_{x_{i}}\mathbf{u}_{k}\cdot\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}$
(3.40)

Subtracting (3.39) from (3.40), and using orthogonality of the projection
(2.42), we obtain

$\displaystyle~{}|\partial_{x_{i}}\mathbf{u}_{k}-\Pi_{\mathbf{u}_{k}}\partial_{x_{i}}\mathbf{u}_{k}|^{2}$
$\displaystyle=$
$\displaystyle~{}|\partial_{x_{i}}\mathbf{u}_{k}|^{2}-|\Pi_{\mathbf{u}_{k}}\partial_{x_{i}}\mathbf{u}_{k}|^{2}$
$\displaystyle\geq$
$\displaystyle~{}(1-C\delta_{0})|\partial_{x_{i}}P_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}+\mathrm{d}_{\mathfrak{m}}^{2}(\mathbf{u}_{k})|\partial_{x_{i}}\partial\mathrm{d}_{\mathfrak{m}}(\mathbf{u}_{k})|^{2}.$

This together with (3.1b) implies (3.33).

∎

######  Theorem 3.6.

The limit $\mathbf{u}^{\pm}$ in (3.17) are weak solutions of harmonic heat
flows from $\cup_{t\in[0,T]}\Omega_{t}^{\pm}$ to $\mathfrak{m}_{\pm}$
respectively and satisfy, in addition to (3.17), that

$\displaystyle\mathbf{u}^{\pm}\in
L^{2}(0,T;H^{1}(\Omega_{t}^{\pm},\mathfrak{m}_{\pm}))$ (3.41)

and . Moreover, with the notations in Proposition 3.5, for a.e. $t\in[0,T]$
there holds

$\displaystyle\mathbf{v}_{k}(\cdot,t):=\sum_{\pm}P_{\mathfrak{m}_{\pm}}\circ\mathbf{u}_{k}(\cdot,t)~{}\mathbf{1}_{\Omega_{t}^{k,\pm}}$
$\displaystyle\xrightarrow{k\to\infty}\mathbf{u}=\sum_{\pm}\mathbf{u}^{\pm}(\cdot,t)~{}\mathbf{1}_{\Omega_{t}^{\pm}}\text{
weakly-star in }BV(\Omega),$ (3.42) $\displaystyle\nabla^{a}\mathbf{v}_{k}$
$\displaystyle\xrightarrow{k\to\infty}\mathbf{1}_{\Omega_{t}^{\pm}}\nabla\mathbf{u}^{\pm}\text{
weakly in }L^{1}(\Omega),$ (3.43)
$\displaystyle\sum_{\pm}\int_{\Omega_{t}^{\pm}}|\nabla\mathbf{u}^{\pm}(\cdot,t)|^{2}\,dx$
$\displaystyle\leq\liminf_{k\to\infty}\sum_{\pm}\int_{\Omega_{t}^{k,\pm}}\left|\nabla^{a}\mathbf{v}_{k}(\cdot,t)\right|^{2}\,dx,$
(3.44)

where $\nabla^{a}\mathbf{v}_{k}$ is the absolute part of the distributional
gradient $\nabla$.

###### Proof.

The sequence $\mathbf{v}_{k}(\cdot,t)$ is bounded in $L^{\infty}(\Omega)$, and
by (3.33) we deduce that their distributional derivatives have no Cantor
parts. Moreover, the absolute continuous parts and the jump sets enjoy the
estimates (3.33) and (3.30) respectively. So it follows from Proposition 2.3
that $\\{\mathbf{v}_{k}(\cdot,t)\\}$ is compact in $SBV(\Omega)$: there exists
$\mathbf{v}\in SBV(\Omega)$ so that $\mathbf{v}_{k}\to\mathbf{v}$ weakly-star
in $BV(\Omega)$ as $k\to\infty$, and the absolute continuous part of the
gradient

$\nabla^{a}\mathbf{v}_{k}=\sum_{\pm}\nabla
P_{\mathfrak{m}_{\pm}}(\mathbf{u}_{k})~{}\mathbf{1}_{\Omega_{t}^{k,\pm}}\xrightarrow{k\to\infty}\nabla^{a}\mathbf{v}\text{
weakly in }L^{1}(\Omega).$ (3.45)

To identify $\mathbf{v}$, we combine (3.31) with (3.18c) and deduce that
$\mathbf{v}=\sum_{\pm}\mathbf{1}_{\Omega_{t}^{+}}\mathbf{u}^{\pm}$ a.e., and
thus (3.42) is proved. By lower semicontinuity of SBV functions (cf. (2.2)),
we get (3.44), and thus we can improve the spatial regularity in (3.17) to
(3.41). Finally combining (3.2), (3.18) with Chen–Struwe [8] implies that,
$\mathbf{u}^{\pm}:\cup_{t\in[0,T]}\Omega_{t}^{\pm}\mapsto\mathfrak{m}_{\pm}$
are weak solutions to harmonic heat flows respectively. ∎

##  4\. Proof of Theorem 1.1

We first recall that the estimate (1.16) is proved in Lemma 3.1 (cf. (3.1a)).
The estimate (1.17) is obtained in Theorem 3.2. Now we prove (1.18): for every
$t\in[0,T]$,

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}\left(\frac{\varepsilon}{2}\left|\nabla\mathbf{u}_{\varepsilon}\right|^{2}+\frac{F(\mathbf{u}_{\varepsilon})}{\varepsilon}\right)dx$
$\displaystyle\overset{\eqref{calibration est2}}{=}\lim_{\varepsilon\to
0}\int_{\Omega}\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx$
$\displaystyle\overset{\eqref{bc n and H}}{=}\lim_{\varepsilon\to
0}-\int_{\Omega}\operatorname{div}\bm{\xi}\,\psi_{\varepsilon}\,dx$
$\displaystyle\overset{\eqref{volume
convergence}}{=}c_{F}\int_{\Omega_{t}^{+}}\operatorname{div}\bm{\xi}\overset{\eqref{def:xi}}{=}c_{F}\mathcal{H}^{d-1}(\Sigma_{t}),$
(4.1)

where we used the divergence theorem in the last step. The convergence (1.19)
is obtained (along with others) in Corollary 3.3. The limit $\mathbf{u}^{\pm}$
being harmonic heat flow with regularity (1.20) has been done in Theorem 3.6.
It remains to prove minimal pair boundary conditions (1.21).

We shall argue for a.e. $t\in[0,T]$. By (3.41) and the trace theorem, for any
$\tau\in[0,\delta_{0})$, we have
$\mathbf{u}^{\pm}(p\pm\tau\mathbf{n}_{\Sigma}(p))\in\mathfrak{m}_{\pm}$ where
$\mathcal{C}_{F}(\cdot,\cdot)$ is well-defined and Lipschitz continuous in
$\mathfrak{m}_{+}\times\mathfrak{m}_{-}$. By (3.41) and Sobolev’s trace
estimate, we have for a.e. $s\in(0,\delta_{0})$ that

$\mathbf{u}^{\pm}\left(p\pm s\mathbf{n}_{\Sigma}(p)\right)\xrightarrow{s\to
0}\mathbf{u}^{\pm}(p)\text{ strongly in }L^{2}(\Sigma_{t}).$ (4.2)

This combined with Lipschitz continuity of $\mathcal{C}_{F}$ yields for a.e.
$s\in(0,\delta_{0})$,

$\mathcal{C}_{F}\left(\mathbf{u}^{+}(p+s\mathbf{n}_{\Sigma}(p)),\mathbf{u}^{-}(p-s\mathbf{n}_{\Sigma}(p))\right)\xrightarrow{s\to
0}\mathcal{C}_{F}(\mathbf{u}^{+}(p),\mathbf{u}^{-}(p))\text{ strongly in
}L^{2}(\Sigma_{t}).$ (4.3)

On the other hand, (3.18c) implies the strong convergence of
$\mathbf{u}_{i}=\mathbf{u}_{\varepsilon_{i}}$ on almost every slices. More
precisely, for a.e. $s\in(0,\delta_{0})$, there holds

$\mathbf{u}_{k}\left(p\pm
s\mathbf{n}_{\Sigma}(p)\right)\xrightarrow{k\to\infty}\mathbf{u}^{\pm}(p\pm
s\mathbf{n}_{\Sigma}(p))\text{ strongly in }L^{2}(\Sigma_{t}).$ (4.4)

Now we shall derive a lower bound of the Ginzburg–Landau energy on the
$s$-neighborhood of $\Sigma_{t}$. By a change of variables
$z=\varepsilon_{k}y$ we have

$\displaystyle\int_{\Omega}\left(\frac{\varepsilon_{k}}{2}\left|\nabla\mathbf{u}_{k}\right|^{2}+\varepsilon_{k}^{-1}F(\mathbf{u}_{k})\right)dx$
$\displaystyle\geq$
$\displaystyle\int_{\Sigma_{t}}\int_{-s}^{s}\left(\frac{\varepsilon_{k}}{2}|\partial_{s}\mathbf{u}_{k}(p+z\mathbf{n}_{\Sigma}(p)))|^{2}+\varepsilon_{k}^{-1}F\left(\mathbf{u}_{k}(p+z\mathbf{n}_{\Sigma}(p))\right)\right)dz\,d\mathcal{H}^{d-1}(p)$
$\displaystyle=$
$\displaystyle\int_{\Sigma_{t}}\int_{-\frac{s}{\varepsilon_{k}}}^{\frac{s}{\varepsilon_{k}}}\left(\frac{1}{2}|\partial_{y}\mathbf{u}_{k}(p+\varepsilon_{k}y\mathbf{n}_{\Sigma}(p))|^{2}+F\left(\mathbf{u}_{k}(p+\varepsilon_{k}y\mathbf{n}_{\Sigma}(p))\right)\right)dy\,d\mathcal{H}^{d-1}(p)$
$\displaystyle\geq$
$\displaystyle\int_{\Sigma_{t}}\alpha_{k}(p,s)\,d\mathcal{H}^{d-1}(p),$ (4.5)

where $\alpha_{k}(p,s)$ (with $p\in\Sigma_{t},$ and a.e.
$s\in(-\delta_{0},0)\cup(0,\delta_{0})$) is defined by

$\displaystyle\alpha_{k}(p,s):=\inf$
$\displaystyle\left\\{\int_{-\frac{s}{\varepsilon_{k}}}^{\frac{s}{\varepsilon_{k}}}\left(\frac{1}{2}|\bm{\gamma}^{\prime}(y)|^{2}+F(\bm{\gamma}(y))\right)dy:\right.$
$\displaystyle\left.\bm{\gamma}\in
H^{1}\left(\left(-\frac{s}{\varepsilon_{k}},\frac{s}{\varepsilon_{k}}\right),\mathbb{R}^{n}\right),\bm{\gamma}\left(\pm\frac{s}{\varepsilon_{k}}\right)=\mathbf{u}_{k}\left(p\pm
s\mathbf{n}_{\Sigma}(p)\right)\right\\}.$

By (4.4) and (2.5), for a.e $s\in(0,\delta_{0})$ we have

$\alpha_{k}(p,s)\xrightarrow{k\to\infty}\mathcal{C}_{F}(\mathbf{u}^{+}(p+s\mathbf{n}_{\Sigma}(p)),\mathbf{u}^{-}(p-s\mathbf{n}_{\Sigma}(p)))\text{
for }\mathcal{H}^{d-1}-a.e.\quad p\in\Sigma_{t}.$ (4.6)

Combining this with (4.3) and a diagonal process, we can pass to a
$s_{k}\downarrow 0$ so that

$\alpha_{k}(p,s_{k})\xrightarrow{k\to\infty}\mathcal{C}_{F}(\mathbf{u}^{+}(p),\mathbf{u}^{-}(p))\text{
for }\mathcal{H}^{d-1}-a.e.\quad p\in\Sigma_{t}.$ (4.7)

Now we turn to the estimate of the Ginzburg–Landau energy: combining (4) with
(4) yields

$\displaystyle c_{F}\mathcal{H}^{d-1}(\Sigma_{t})$
$\displaystyle\overset{\eqref{calibration
est3}}{=}\liminf_{k\to\infty}\int_{\Omega}\left(\frac{\varepsilon_{k}}{2}\left|\nabla\mathbf{u}_{k}\right|^{2}+\varepsilon_{k}^{-1}F(\mathbf{u}_{k})\right)dx$
$\displaystyle\overset{\eqref{der
mp1}}{\geq}\liminf_{k\to\infty}\int_{\Sigma_{t}}\alpha_{k}(p,s_{k})\,d\mathcal{H}^{d-1}(p)$
$\displaystyle\overset{\eqref{alphaconv1}}{\geq}\int_{\Sigma_{t}}\mathcal{C}_{F}(\mathbf{u}^{+}(p),\mathbf{u}^{-}(p))\,d\mathcal{H}^{d-1}(p).$
(4.8)

Note that in the last step we used (4.7) and the Fatou’s lemma. By Lemma 2.6,
we have $\mathcal{C}_{F}(\mathbf{u}^{+}(p),\mathbf{u}^{-}(p))\geq c_{F}$ for
$\mathcal{H}^{d-1}$-a.e. $p\in\Sigma_{t}$, and by (4) we actually obtain
equality here. So the minimal pair boundary conditions (1.21) follows from
(2.13).

##  5\. Construction of initial data: Proof of Theorem 1.2

We shall first modify and extend $\mathbf{u}^{in}_{\pm}$ in the transitional
region $B_{\delta}(\Sigma_{0})$ so that we can glue them into a new mapping
that fulfills the desired properties in Theorem 1.2. Let
$\Psi_{\delta}:\Omega_{0}^{\pm}\backslash
B_{\delta}(\Sigma_{0})\mapsto\Omega_{0}^{\pm}$ be global diffeomorphism up to
the boundaries so that

$\displaystyle\Psi_{\delta}(p\pm\delta\mathbf{n}_{\Sigma_{0}}(p))$
$\displaystyle=p,\quad\forall p\in\Sigma_{0},$ (5.1)
$\displaystyle\Psi_{\delta}(x)$ $\displaystyle=x,\quad\forall
x\in\Omega_{0}^{\pm}\backslash B_{2\delta}(\Sigma_{0}).$ (5.2)

We extend $\mathbf{u}_{in}^{\pm}$ to some $\mathbf{u}_{0}^{\pm}\in
H^{1}(\Omega_{0}^{\pm}\cup\overline{B_{\delta}(\Sigma_{0})},\mathfrak{m}_{\pm})$
defined by

$\mathbf{u}_{0}^{\pm}=\left\\{\begin{split}\mathbf{u}_{in}^{\pm}\circ\Psi_{\delta}&\text{
in }&\Omega_{0}^{\pm}\backslash B_{\delta}(\Sigma_{0}),\\\
\mathbf{u}_{in}^{\pm}\circ P_{\Sigma_{0}}&\text{ in
}&\overline{B_{\delta}(\Sigma_{0})}.\end{split}\right.$ (5.3)

Note that $\mathbf{u}_{0}^{\pm}$ in (5.3) are constant in normal modifications
of $\mathbf{u}_{in}^{\pm}$ in $B_{\delta}(\Sigma_{0})$. This combined with
(1.25) yields

$\displaystyle(\mathbf{u}_{0}^{+}(x),\mathbf{u}_{0}^{-}(x))|_{B_{\delta}(\Sigma_{0})}\text{
being minimal pairs}.$ (5.4)

We shall construct $\mathbf{u}_{\varepsilon}^{in}$ by gluing
$\mathbf{u}_{0}^{\pm}$. To this end, we define a cut-off function

$\eta_{\delta}\in C_{c}^{\infty}(B_{\delta}(\Sigma_{t});[0,1])~{}\text{ with
}\eta_{\delta}=1\text{ in }B_{\delta/2}(\Sigma_{t}).$ (5.5)

Recall the optimal profile $\alpha$ (2.10) and the cut-off function (2.31). We
define

$S_{\varepsilon}(x)=\eta_{\delta}(x,0)\alpha\left(\tfrac{\mathrm{d}_{\Sigma}(x,0)}{\varepsilon}\right)+(1-\eta_{\delta}(x,0))\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\left(\mathbf{1}_{\Omega^{+}_{0}}(x)-\mathbf{1}_{\Omega^{-}_{0}}(x)\right).$
(5.6)

Note that the discontinuity caused by $\mathbf{1}_{\Omega^{\pm}_{0}}$ is cut
off by $\eta_{\delta}$. So $S_{\varepsilon}$ is a smooth function, and we
extract its leading order by

$S_{\varepsilon}(x)=\alpha\left(\tfrac{\mathrm{d}_{\Sigma}(x,0)}{\varepsilon}\right)-\hat{S}_{\varepsilon}(x),$
(5.7)

where $\hat{S}_{\varepsilon}$ is the tail term

$\hat{S}_{\varepsilon}(x)=(1-\eta_{\delta}(x,0))\left(\alpha\left(\tfrac{\mathrm{d}_{\Sigma}(x,0)}{\varepsilon}\right)-\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\left(\mathbf{1}_{\Omega^{+}_{0}}(x)-\mathbf{1}_{\Omega^{-}_{0}}(x)\right)\right).$
(5.8)

We note that $\mathrm{d}_{\Sigma}(x,0)$ is Lipschitz continuous in $\Omega$,
and by Rademacher’s theorem $|\nabla\mathrm{d}_{\Sigma}(x,0)|\leq 1$ a.e. in
$\Omega$. This combined with (2.10d) yields

$\|\hat{S}_{\varepsilon}\|_{L^{\infty}(\Omega)}+\|\nabla\hat{S}_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq
Ce^{-C/\varepsilon}.$ (5.9)

and thus

$\displaystyle S_{\varepsilon}(x)$
$\displaystyle=\alpha\left(\tfrac{\mathrm{d}_{\Sigma}(x,0)}{\varepsilon}\right)+O(e^{-C/\varepsilon})\text{
in }\Omega,$ (5.10a) $\displaystyle\nabla S_{\varepsilon}(x)$
$\displaystyle=\tfrac{\nabla\mathrm{d}_{\Sigma}(x,0)}{\varepsilon}\alpha^{\prime}+O(e^{-C/\varepsilon})\quad\text{a.e.
in }\Omega,$ (5.10b) $\displaystyle S_{\varepsilon}(x)$
$\displaystyle\xrightarrow{\varepsilon\to
0}\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}\left(\mathbf{1}_{\Omega^{+}_{0}}(x)-\mathbf{1}_{\Omega^{-}_{0}}(x)\right)\text{
a.e. in }\Omega.$ (5.10c)

Using (5.3) and (5.6), we define $\mathbf{u}_{\varepsilon}^{in}$ by

$\displaystyle\mathbf{u}_{\varepsilon}^{in}(x)=\tfrac{\mathbf{u}_{0}^{+}(x)+\mathbf{u}_{0}^{-}(x)}{2}+S_{\varepsilon}(x)\tfrac{\mathbf{u}_{0}^{+}(x)-\mathbf{u}_{0}^{-}(x)}{\mathrm{dist}_{\mathfrak{m}}}.$
(5.11)

We claim (1.26) holds. Indeed in the domains $\Omega_{0}^{\pm}\backslash
B_{2\delta}(\Sigma_{0})$, we have $\eta_{\delta}=0$ and thus

$\mathbf{u}_{\varepsilon}^{in}\overset{\eqref{se
def}}{=}\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}(\mathbf{1}_{\Omega^{+}_{0}}-\mathbf{1}_{\Omega^{-}_{0}})\tfrac{\mathbf{u}_{0}^{+}(x)-\mathbf{u}_{0}^{-}(x)}{\mathrm{dist}_{\mathfrak{m}}}=\sum_{\pm}\mathbf{u}_{0}^{\pm}\mathbf{1}_{\Omega^{\pm}_{0}}.$
(5.12)

This combined with (5.3) yields

$\mathbf{u}_{\varepsilon}^{in}(x)\overset{\eqref{u in
extension}}{=}\mathbf{u}_{in}^{\pm}\circ\Psi_{\delta}(x)\overset{\eqref{psi
fix}}{=}\mathbf{u}_{in}^{\pm}(x),\quad\forall x\in\Omega_{0}^{\pm}\backslash
B_{2\delta}(\Sigma_{0}).$ (5.13)

Now we turn to the verification of (1.27). Substituting (5.10a) and (5.10b)
into (5.11), we obtain

$\displaystyle\nabla\mathbf{u}_{\varepsilon}^{in}$
$\displaystyle=\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\tfrac{\nabla\mathrm{d}_{\Sigma}}{\varepsilon}\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}$
$\displaystyle+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}+O(e^{-C/\varepsilon})\left(|\nabla\mathbf{u}_{0}^{+}|+|\nabla\mathbf{u}_{0}^{-}|\right)\qquad\text{a.e.
in }\Omega.$ (5.14)

###  5.1. Proof of (1.27): Estimate near $\Sigma_{0}$

By (5.4) and the second part of Lemma 2.6, we have

$(\mathbf{u}_{0}^{+}(x)-\mathbf{u}_{0}^{-}(x))\perp_{\mathbb{R}^{n}}T_{\mathbf{u}_{0}^{\pm}(x)}\mathfrak{m}_{\pm},\quad\forall
x\in B_{\delta}(\Sigma_{0}).$ (5.15)

Since $\mathbf{u}_{0}^{\pm}$ map into $\mathfrak{m}_{\pm}$ respectively, we
have $\nabla\mathbf{u}_{0}^{\pm}(x)\in
T_{\mathbf{u}^{\pm}(x)}\mathfrak{m}_{\pm}$. So we have for $1\leq i\leq k$
that

$(\mathbf{u}_{0}^{+}(x)-\mathbf{u}_{0}^{-}(x))\cdot\partial_{x_{i}}\mathbf{u}_{0}^{\pm}(x),\quad\forall
x\in B_{\delta}(\Sigma_{0}).$ (5.16)

Using this equation we can compute the square of (5): the square of the second
term on the right hand side of (5) is

$\left|\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon}\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|\overset{\eqref{initial
mp}}{=}\varepsilon^{-2}\left(\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)^{2}\text{
in }B_{\delta}(\Sigma_{0}).$ (5.17)

This together with (5.16) enable us to compute the square of (5) by

$\displaystyle|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}=$
$\displaystyle\varepsilon^{-2}\left(\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)^{2}+\left|\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|^{2}$
$\displaystyle+O(e^{-C/\varepsilon})\left(|\nabla\mathbf{u}_{0}^{+}|+|\nabla\mathbf{u}_{0}^{-}|\right)^{2}\qquad\text{
in }B_{\delta}(\Sigma_{0}).$ (5.18)

By (5.10a) and (5.11), we have

$F(\mathbf{u}_{\varepsilon}^{in})=F\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)+O(e^{-C/\varepsilon})\qquad\text{
in }\Omega.$ (5.19)

To be more explicit on the RHS of (5.19), we first deduce from (2.10) that

$\bm{\gamma}(s):\mathbb{R}\mapsto\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha\left(s\right)\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}$
(5.20)

prescribes the line segment $\overline{\mathbf{u}_{0}^{-}\mathbf{u}_{0}^{+}}$
and $\gamma(0)=\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}$ is the middle
point. For $x\in B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{+}$, we have
$\alpha(\tfrac{\mathrm{d}_{\Sigma}(x)}{\varepsilon})>0$ and by (2.10a), the
distance from $\bm{\gamma}(\frac{\mathrm{d}_{\Sigma}(x)}{\varepsilon})$ to
$\mathfrak{m}$ equals to the distance between
$\bm{\gamma}(\frac{\mathrm{d}_{\Sigma}(x)}{\varepsilon})$ and
$\mathbf{u}_{0}^{+}\in\mathfrak{m}_{+}$. So

$\left|\mathrm{d}_{\mathfrak{m}}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)\right|=\left|\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}-\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right|\left|\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|\overset{\eqref{initial
mp}}{=}\left|\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}-\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right|$
(5.21)

on $B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{+}$, and thus

$\displaystyle
F\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)$
$\displaystyle\overset{\eqref{bulk potential},\eqref{dist repre}}{=}$
$\displaystyle
f\left(\left|\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}-\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right|^{2}\right)\overset{\eqref{centralized
potential}}{=}\widetilde{F}\left(\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right).$
(5.22)

Similar calculation gives the case when $x\in
B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{-}$, and to summarize we have from
(5.19) and (5.1) that

$F(\mathbf{u}_{\varepsilon}^{in})=\widetilde{F}\left(\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)+O(e^{-C/\varepsilon})\qquad\text{
in }B_{\delta}(\Sigma_{t}).$ (5.23)

Now we compute
$\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon})$:

$\displaystyle-\int_{\Omega}\eta_{\delta}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})$
$\displaystyle\overset{\eqref{cut-off eta
delta2}}{=}\int_{\Omega}\operatorname{div}(\eta_{\delta}\bm{\xi})\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}$
$\displaystyle\overset{\eqref{uu
initial}}{=}\int_{\Omega}\operatorname{div}(\eta_{\delta}\bm{\xi})\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+S_{\varepsilon}\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)$
(5.24)

Using (5.10a) and the Lipschitz property of $\mathrm{d}_{F}$, we can write

$\displaystyle-\int_{\Omega}\eta_{\delta}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})=$
$\displaystyle\int_{\Omega}\operatorname{div}(\eta_{\delta}\bm{\xi})\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)+O(e^{-C/\varepsilon}).$
(5.25)

Thanks to (5.4), we can find the value of $\mathrm{d}_{F}$ on
$B_{\delta}(\Sigma_{t})$: there are four cases in the definition (2.18) for
$\mathrm{d}_{F}$, but over a line segment inside a minimal connection involves
only the second and the fourth cases. To be more precise, for any $x\in
B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{+}$, since
$\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})>0$ and
$(\mathbf{u}_{0}^{+},\mathbf{u}_{0}^{-})$ are minimal pairs (cf. (5.4)), we
have

$\displaystyle\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)$
$\displaystyle\overset{\eqref{quasidistance}}{=}c_{F}-\int_{0}^{\mathrm{d}_{\mathfrak{m}_{+}}\big{(}\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\big{)}}\sqrt{2f(\lambda^{2})}\,d\lambda$
$\displaystyle=c_{F}-\int_{0}^{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}-\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}\sqrt{2f(\lambda^{2})}\,d\lambda$
$\displaystyle\overset{\eqref{centralized
potential}}{=}c_{F}-\int_{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}^{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\qquad\forall
x\in B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{+}.$

Similar calculation applies to the case when $x\in
B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{-}$ and
$\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})<0$:

$\displaystyle\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)$
$\displaystyle\overset{\eqref{quasidistance}}{=}\int_{0}^{\mathrm{d}_{\mathfrak{m}_{-}}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)}\sqrt{2f(\lambda^{2})}\,d\lambda$
$\displaystyle\overset{\eqref{linpanwang cf
equ}}{=}\int_{0}^{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}\sqrt{2f(\lambda^{2})}\,d\lambda$
$\displaystyle\overset{\eqref{centralized
potential}}{=}\int_{-\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}^{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\qquad\forall
x\in B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{-}.$

To summarize, we have

$\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)=\left\\{\begin{split}c_{F}-\int_{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}^{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\qquad\forall
x\in B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{+},\\\
\int_{-\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}^{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\qquad\forall
x\in B_{\delta}(\Sigma_{0})\cap\Omega_{0}^{-}.\end{split}\right.$ (5.26)

Recall from (5.5) that $\eta_{\delta}$ vanishes outside
$B_{\delta}(\Sigma_{0})$. So substituting (5.26) into (5.25) and integrating
by parts yield

$\displaystyle\int_{\Omega}\eta_{\delta}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})=$
$\displaystyle\int_{\Omega}\varepsilon^{-1}\eta_{\delta}\bm{\xi}\cdot\nabla\mathrm{d}_{\Sigma}\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\sqrt{2\widetilde{F}\left(\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)}\,dx+O(e^{-C/\varepsilon}).$
(5.27)

This combined with (5.23) and (5.1) yields

$\displaystyle\int_{\Omega}\left(\frac{1}{2}|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}+\frac{F(\mathbf{u}_{\varepsilon}^{in})}{\varepsilon^{2}}-\frac{1}{\varepsilon}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})\right)\eta_{\delta}$
$\displaystyle=\int_{\Omega}\left(\varepsilon^{-2}\frac{1}{2}\left(\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)^{2}+\varepsilon^{-2}\widetilde{F}\left(\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)-\varepsilon^{-2}\bm{\xi}\cdot\nabla\mathrm{d}_{\Sigma}\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\sqrt{2\widetilde{F}\left(\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)}\right)\eta_{\delta}$
$\displaystyle+\int_{\Omega}\frac{1}{2}\eta_{\delta}\left|\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|^{2}$
$\displaystyle+O(e^{-C/\varepsilon})\int_{\Omega}\left(1+|\nabla\mathbf{u}_{0}^{+}|^{2}+|\nabla\mathbf{u}_{0}^{-}|^{2}\right)\eta_{\delta}.$
(5.28)

By (2.10c) the first term on the right hand side above simplifies to

$\displaystyle\int_{\Omega}\eta_{\delta}\varepsilon^{-2}(1-\bm{\xi}\cdot\nabla\mathrm{d}_{\Sigma})\left(\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)^{2}\overset{\eqref{def:xi}}{=}\int_{\Omega}O(1)\eta_{\delta}\tfrac{\mathrm{d}_{\Sigma}^{2}}{\varepsilon^{2}}\left(\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\right)^{2}\overset{\eqref{exp
alpha}}{=}O(\varepsilon).$ (5.29)

The above two equations together implies

$\displaystyle\int_{\Omega}\left(\frac{1}{2}|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}+\frac{F(\mathbf{u}_{\varepsilon}^{in})}{\varepsilon^{2}}-\frac{1}{\varepsilon}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})\right)\eta_{\delta}$
$\displaystyle=\int_{\Omega}\frac{1}{2}\eta_{\delta}\left|\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|^{2}+O(\varepsilon)\int_{\Omega}\left(1+|\nabla\mathbf{u}_{0}^{+}|^{2}+|\nabla\mathbf{u}_{0}^{-}|^{2}\right)\eta_{\delta}.$
(5.30)

###  5.2. Proof of (1.27): Estimates away from $\Sigma_{0}$.

Using (2.10d), we have

$|\alpha^{\prime}(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})|+\left|\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})-\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}(\mathbf{1}_{\Omega_{0}^{+}}-\mathbf{1}_{\Omega_{0}^{-}})\right|\leq
Ce^{-C/\varepsilon}\text{ in }\Omega\backslash B_{\delta/2}(\Sigma_{0}).$
(5.31)

Applying the above estimates to (5) yields

$\displaystyle\nabla\mathbf{u}_{\varepsilon}^{in}$
$\displaystyle=\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}(\mathbf{1}_{\Omega_{0}^{+}}-\mathbf{1}_{\Omega_{0}^{-}})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}$
$\displaystyle\qquad+O(e^{-C/\varepsilon})\left(1+|\nabla\mathbf{u}_{0}^{+}|\mathbf{1}_{\Omega_{0}^{+}}+|\nabla\mathbf{u}_{0}^{-}|\mathbf{1}_{\Omega_{0}^{-}}\right)$
$\displaystyle=\nabla\mathbf{u}_{0}^{+}\mathbf{1}_{\Omega_{0}^{+}}+\nabla\mathbf{u}_{0}^{-}\mathbf{1}_{\Omega_{0}^{-}}$
$\displaystyle\qquad+O(e^{-C/\varepsilon})\left(1+|\nabla\mathbf{u}_{0}^{+}|\mathbf{1}_{\Omega_{0}^{+}}+|\nabla\mathbf{u}_{0}^{-}|\mathbf{1}_{\Omega_{0}^{-}}\right)\quad\text{a.e.
in }\Omega\backslash B_{\delta/2}(\Sigma_{0}).$ (5.32)

By (5.5) the function $(1-\eta_{\delta})$ vanishes on
$B_{\delta/2}(\Sigma_{0}).$ So multiplying this function to (5.2) yields

$\displaystyle(1-\eta_{\delta})|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}$
$\displaystyle=(1-\eta_{\delta})\sum_{\pm}|\nabla\mathbf{u}_{0}^{\pm}|^{2}\mathbf{1}_{\Omega_{0}^{\pm}}$
$\displaystyle\quad+O(e^{-C/\varepsilon})\left(1+\sum_{\pm}|\nabla\mathbf{u}_{0}^{\pm}|^{2}\mathbf{1}_{\Omega_{0}^{\pm}}\right)\qquad\text{a.e.
in }\Omega.$ (5.33)

Similar but easier calculation of (5.19) yields

$\displaystyle(1-\eta_{\delta})F(\mathbf{u}_{\varepsilon}^{in})$
$\displaystyle=O(e^{-C/\varepsilon})\qquad\text{ in }\Omega.$ (5.34)

By the Lipschitz continuity of $\mathrm{d}_{F}$, (5.11) and (5.31), we have on
$\Omega\backslash B_{\delta/2}(\Sigma_{0})$ that

$\displaystyle\mathrm{d}_{F}(\mathbf{u}_{\varepsilon}^{in})$
$\displaystyle\overset{\eqref{uu
initial}}{=}\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+S_{\varepsilon}\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)$
$\displaystyle\overset{\eqref{def
Se1}}{=}\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)+O(e^{-C/\varepsilon})$
$\displaystyle\overset{\eqref{exp
alpha1}}{=}\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}(\mathbf{1}_{\Omega_{0}^{+}}-\mathbf{1}_{\Omega_{0}^{-}})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)+O(e^{-C/\varepsilon})$
$\displaystyle\,~{}=\mathrm{d}_{F}\left(\mathbf{u}_{0}^{+}\mathbf{1}_{\Omega_{0}^{+}}+\mathbf{u}_{0}^{-}\mathbf{1}_{\Omega_{0}^{-}}\right)+O(e^{-C/\varepsilon})$

As $\mathbf{u}_{0}^{\pm}$ map into $\mathfrak{m}_{\pm}$ (cf. (5.3)), we have

$\displaystyle\mathrm{d}_{F}(\mathbf{u}_{\varepsilon}^{in})$
$\displaystyle\overset{\eqref{quasidistance}}{=}c_{F}\mathbf{1}_{\Omega_{0}^{+}}+O(e^{-C/\varepsilon})\quad\text{
in }\Omega\backslash B_{\delta/2}(\Sigma_{0}).$ (5.35)

As $(1-\eta_{\delta})$ vanishes on $B_{\delta/2}(\Sigma_{0})$,

$\displaystyle\operatorname{div}\left((1-\eta_{\delta})\bm{\xi}\right)\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}\overset{\eqref{quasidistance}}{=}\operatorname{div}\left((1-\eta_{\delta})\bm{\xi}\right)c_{F}\mathbf{1}_{\Omega_{0}^{+}}+O(e^{-C/\varepsilon}).$
(5.36)

where $c_{F}$ is the constant (2.6). So we have

$\displaystyle-$
$\displaystyle\int_{\Omega}(1-\eta_{\delta})\bm{\xi}\cdot\nabla\left(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}\right)$
$\displaystyle\overset{\eqref{bc n and
H}}{=}\int_{\Omega}\operatorname{div}\left((1-\eta_{\delta})\bm{\xi}\right)\mathrm{d}_{F}(\mathbf{u}_{\varepsilon}^{in})$
$\displaystyle\overset{\eqref{grad
uuin8}}{=}\int_{\Omega_{0}^{+}}\operatorname{div}\left((1-\eta_{\delta})\bm{\xi}\right)c_{F}+O(e^{-C/\varepsilon})$
$\displaystyle\overset{\eqref{grad
uuin8}}{=}c_{F}\int_{\Sigma_{0}}(1-\eta_{\delta})\bm{\xi}\cdot\nu\,d\mathcal{H}^{d-1}+O(e^{-C/\varepsilon})=O(e^{-C/\varepsilon}).$
(5.37)

Putting (5.2), (5.34) and (5.2) together, we obtain

$\displaystyle\int_{\Omega}\left(\frac{1}{2}|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}+\frac{F(\mathbf{u}_{\varepsilon}^{in})}{\varepsilon^{2}}-\frac{1}{\varepsilon}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})\right)(1-\eta_{\delta})$
$\displaystyle=\sum_{\pm}\int_{\Omega_{0}^{\pm}}\frac{1-\eta_{\delta}}{2}|\nabla\mathbf{u}_{0}^{\pm}|^{2}+O(e^{-C/\varepsilon})+O(e^{-C/\varepsilon})\sum_{\pm}\int_{\Omega_{0}^{\pm}}|\nabla\mathbf{u}_{0}^{\pm}|^{2}.$
(5.38)

Combining this with (5.1) leads to

$\displaystyle\int_{\Omega}\left(\frac{1}{2}|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}+\frac{F(\mathbf{u}_{\varepsilon}^{in})}{\varepsilon^{2}}-\frac{1}{\varepsilon}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})\right)$
$\displaystyle=\int_{\Omega}\frac{1}{2}\eta_{\delta}\left|\tfrac{\nabla\mathbf{u}_{0}^{+}+\nabla\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\nabla\mathbf{u}_{0}^{+}-\nabla\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right|^{2}+\sum_{\pm}\int_{\Omega_{0}^{\pm}}\frac{1-\eta_{\delta}}{2}|\nabla\mathbf{u}_{0}^{\pm}|^{2}+O(\varepsilon).$
(5.39)

This leads to (1.27).

As a byproduct, we compute the limit of (5.2). As
$\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\xrightarrow{\varepsilon\to
0}\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}(\mathbf{1}_{\Omega_{0}^{+}}-\mathbf{1}_{\Omega_{0}^{-}})$
a.e. in $\Omega$, we can apply the dominated convergence to the first integral
on the RHS of (5.2) and get

$\displaystyle\lim_{\varepsilon\to
0}\int\left(\frac{1}{2}|\nabla\mathbf{u}_{\varepsilon}^{in}|^{2}+\frac{F(\mathbf{u}_{\varepsilon}^{in})}{\varepsilon^{2}}-\frac{1}{\varepsilon}\bm{\xi}\cdot\nabla(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in})\right)=\sum_{\pm}\int_{\Omega_{0}^{\pm}}|\nabla\mathbf{u}_{0}^{\pm}|^{2}.$
(5.40)

###  5.3. Proof of (1.28).

Recall (1.14) that

$\displaystyle B[\mathbf{u}_{\varepsilon}^{in}|\Sigma_{0}]:=$
$\displaystyle\int_{\Omega}\Big{(}c_{F}\chi-
c_{F}+2\left(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}-c_{F}\right)^{-}\Big{)}\eta\circ\mathrm{d}_{\Sigma}\,dx$
$\displaystyle+\int_{\Omega}\left(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}-c_{F}\right)^{+}|\eta\circ\mathrm{d}_{\Sigma}|\,dx,$
(5.41)

where $\chi=\mathbf{1}_{\Omega_{0}^{+}}-\mathbf{1}_{\Omega_{0}^{-}}$, and
$\eta$ is defined by (3.8). We also recall from (1.13) and (2.9) that

$c_{F}=2\int_{0}^{\tfrac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}d\lambda.$
(5.42)

Using (5.10a) and the Lipschitz property of $\mathrm{d}_{F}$,

$\displaystyle\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}\overset{\eqref{uu
initial}}{=}\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+S_{\varepsilon}\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)\overset{\eqref{def
Se1}}{=}\mathrm{d}_{F}\left(\tfrac{\mathbf{u}_{0}^{+}+\mathbf{u}_{0}^{-}}{2}+\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})\tfrac{\mathbf{u}_{0}^{+}-\mathbf{u}_{0}^{-}}{\mathrm{dist}_{\mathfrak{m}}}\right)+O(e^{-C/\varepsilon})$
(5.43)

This combined with (5.35) and (5.26) implies that the second integral of
(5.41) is of order $O(e^{-C/\varepsilon})$. Concerning the first one, we first
deduce from (5.35) that its integrand is of order $O(e^{-C/\varepsilon})$ on
$\Omega\backslash B_{\delta/2}(\Sigma_{0})$. So it remains to estimate the
integral of the transitional region $B_{\delta}(\Sigma_{0})$:

$\displaystyle\int_{\Omega_{0}^{+}\cap
B_{\delta}(\Sigma_{0})}\Big{(}c_{F}\chi-
c_{F}+2\left(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}-c_{F}\right)^{-}\Big{)}\eta\circ\mathrm{d}_{\Sigma}\,dx$
$\displaystyle\overset{\eqref{longcase1}}{=}$
$\displaystyle~{}2\varepsilon\int_{\Omega_{0}^{+}\cap
B_{\delta}(\Sigma_{0})}\left(\int_{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}^{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\right)\frac{\eta\circ\mathrm{d}_{\Sigma}}{\varepsilon}\,dx+O(e^{-C/\varepsilon})\leq
C\varepsilon$ (5.44)

by a change of variable. In a similar way

$\displaystyle\int_{\Omega_{0}^{-}\cap
B_{\delta}(\Sigma_{0})}\Big{(}c_{F}\chi-
c_{F}+2\left(\mathrm{d}_{F}\circ\mathbf{u}_{\varepsilon}^{in}-c_{F}\right)^{-}\Big{)}\eta\circ\mathrm{d}_{\Sigma}\,dx$
$\displaystyle\overset{\eqref{longcase1}}{=}$
$\displaystyle~{}-2\varepsilon\int_{\Omega_{0}^{-}\cap
B_{\delta}(\Sigma_{0})}\left(\int^{\alpha(\tfrac{\mathrm{d}_{\Sigma}}{\varepsilon})}_{\frac{\mathrm{dist}_{\mathfrak{m}}}{2}}\sqrt{2\widetilde{F}(\lambda)}\,d\lambda\right)\frac{\eta\circ\mathrm{d}_{\Sigma}}{\varepsilon}\,dx\overset{\eqref{truncation
eta}}{\leq}C\varepsilon.$ (5.45)

All together we finish the proof of (1.28).

Acknowledgements. Y. Liu is partially supported by NSF of China under Grant
11971314.

##  Appendix A Proof of Proposition 2.10

As we shall not integrate the time variable $t$ throughout this section, we
shall abbreviate the spatial integration $\int_{\Omega}$ by $\int$ and
sometimes we omit the $\,dx$. We shall also employ Einstein’s summation
convention by summing over repeated Latin indices.

######  Lemma A.1.

The following identity holds

$\displaystyle\int\nabla\mathbf{H}:(\bm{\xi}\otimes\mathbf{n}_{\varepsilon})\left|\nabla\psi_{\varepsilon}\right|\,dx-\int(\nabla\cdot\mathbf{H})\,\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx$
$\displaystyle=$
$\displaystyle\int\nabla\mathbf{H}:(\bm{\xi}-\mathbf{n}_{\varepsilon})\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|\,dx+\int\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\,dx$
$\displaystyle+\int\nabla\cdot\mathbf{H}\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\right)\,dx+\int\nabla\cdot\mathbf{H}(|\nabla\psi_{\varepsilon}|-\bm{\xi}\cdot\nabla\psi_{\varepsilon})\,dx$
$\displaystyle-\int(\nabla\mathbf{H})_{ij}\,\varepsilon\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}\right)\,dx+\int\nabla\mathbf{H}:(\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon})\left|\nabla\psi_{\varepsilon}\right|\,dx$
(A.1)

###### Proof.

We introduce the energy stress tensor
$(T_{\varepsilon})_{ij}=\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})\right)\delta_{ij}-\varepsilon\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}$.
By (2.41b), we have the identity $\nabla\cdot
T_{\varepsilon}=\mathbf{H}_{\varepsilon}|\nabla\mathbf{u}_{\varepsilon}|.$
Testing this identity by $\mathbf{H}$, integrating by parts and using (2.34),
we obtain

$\begin{split}&\int\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\,dx=-\int\nabla\mathbf{H}\colon
T_{\varepsilon}\,dx,\\\
&=-\int\nabla\cdot\mathbf{H}\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})\right)\,dx+\int(\nabla\mathbf{H})_{ij}\,\varepsilon\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}\right)dx.\end{split}$

So adding zero leads to

$\begin{split}&\int\nabla\mathbf{H}:\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|dx\\\
&=\int\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\,dx+\int\nabla\cdot\mathbf{H}\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\right)\,dx+\int\nabla\cdot\mathbf{H}|\nabla\psi_{\varepsilon}|\,dx\\\
&-\int(\nabla\mathbf{H})_{ij}\,\varepsilon\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}\right)dx+\int(\nabla\mathbf{H}):(\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon})\left|\nabla\psi_{\varepsilon}\right|dx.\end{split}$

This easily implies (A.1). ∎

The second lemma gives the expansion of the time derivative of (1.11).

######  Lemma A.2.

Under the assumptions of Theorem 1.1, the following identity holds

$\displaystyle\frac{d}{dt}E$
$\displaystyle[\mathbf{u}_{\varepsilon}|\Sigma]+\frac{1}{2\varepsilon}\int\left(\varepsilon^{2}\left|\partial_{t}\mathbf{u}_{\varepsilon}\right|^{2}-|\mathbf{H}_{\varepsilon}|^{2}\right)\,dx$
$\displaystyle+\frac{1}{2\varepsilon}\int\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\Big{|}^{2}dx+\frac{1}{2\varepsilon}\int\Big{|}\mathbf{H}_{\varepsilon}-\varepsilon|\nabla\mathbf{u}_{\varepsilon}|\mathbf{H}\Big{|}^{2}\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{2\varepsilon}\int\Big{|}(\nabla\cdot\bm{\xi})|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|\mathbf{n}_{\varepsilon}+\varepsilon|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|\mathbf{H}\Big{|}^{2}\,dx$
(A.2a)
$\displaystyle+\frac{\varepsilon}{2}\int|\mathbf{H}|^{2}\left(|\nabla\mathbf{u}_{\varepsilon}|^{2}-|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}\right)\,dx-\int\nabla\mathbf{H}\cdot(\bm{\xi}-\mathbf{n}_{\varepsilon})^{\otimes
2}\left|\nabla\psi_{\varepsilon}\right|\,dx$ (A.2b)
$\displaystyle+\int\left(\nabla\cdot\mathbf{H}\right)\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\right)\,dx$
(A.2c)
$\displaystyle+\int\left(\nabla\cdot\mathbf{H}\right)\left(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\right)|\nabla\psi_{\varepsilon}|\,dx+\int(J_{\varepsilon}^{1}+J_{\varepsilon}^{2})\,dx,$
(A.2d)

where $J_{\varepsilon}^{1},J_{\varepsilon}^{2}$ are given by

$\displaystyle J_{\varepsilon}^{1}:=$
$\displaystyle\nabla\mathbf{H}:\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}\left(|\nabla\psi_{\varepsilon}|-\varepsilon|\nabla\mathbf{u}_{\varepsilon}|^{2}\right)$
$\displaystyle+\varepsilon\nabla\mathbf{H}:(\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon})\left(|\nabla\mathbf{u}_{\varepsilon}|^{2}-|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}\right)$
$\displaystyle-\sum_{i,j}\varepsilon(\nabla\mathbf{H})_{ij}(\partial_{i}\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon})\cdot(\partial_{j}\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\partial_{j}\mathbf{u}_{\varepsilon})\,$
(A.3) $\displaystyle J_{\varepsilon}^{2}:=$
$\displaystyle-\left(\partial_{t}\bm{\xi}+\left(\mathbf{H}\cdot\nabla\right)\bm{\xi}+\left(\nabla\mathbf{H}\right)^{\mathsf{T}}\bm{\xi}\right)\cdot\nabla\psi_{\varepsilon}$
(A.4)

###### Proof of Lemma A.2.

We shall employ the Einstein summation convention by summing over repeated
Latin indices. Using the energy dissipation law (2.35) and adding zero, we
compute the time derivative of the energy (1.11) by

$\displaystyle\frac{d}{dt}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]+\varepsilon\int|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dx-\int(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{t}\mathbf{u}_{\varepsilon}\,dx$
$\displaystyle=$
$\displaystyle\int\left(\mathbf{H}\cdot\nabla\right)\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx+\int\left(\nabla\mathbf{H}\right)^{\mathsf{T}}\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx+\int
J_{\varepsilon}^{2}\,dx$ (A.5)

Due to the symmetry of the Hessian of $\psi_{\varepsilon}$ and the boundary
conditions (2.34), we have

$\displaystyle\int\nabla\cdot(\bm{\xi}\otimes\mathbf{H})\cdot\nabla\psi_{\varepsilon}\,dx=\int\nabla\cdot(\mathbf{H}\otimes\bm{\xi})\cdot\nabla\psi_{\varepsilon}\,dx.$

Hence, the first integral on the right-hand side of (A.5) can be rewritten as

$\displaystyle\int\left(\mathbf{H}\cdot\nabla\right)\,\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx$
$\displaystyle=\int\nabla\cdot(\bm{\xi}\otimes\mathbf{H})\cdot\nabla\psi_{\varepsilon}\,dx-\int(\nabla\cdot\mathbf{H})\,\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx$
$\displaystyle=\int(\nabla\cdot\bm{\xi})\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx+\int(\bm{\xi}\cdot\nabla)\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx-\int(\nabla\cdot\mathbf{H})\,\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx.$

Therefore

$\begin{split}&\frac{d}{dt}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]+\varepsilon\int|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dx-\int(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{t}\mathbf{u}_{\varepsilon}\,dx\\\
=&\int(\nabla\cdot\bm{\xi})\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}dx+\int(\bm{\xi}\cdot\nabla)\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx-\int(\nabla\cdot\mathbf{H})\,\bm{\xi}\cdot\nabla\psi_{\varepsilon}\,dx\\\
&+\int\nabla\mathbf{H}:\left(\bm{\xi}\otimes\mathbf{n}_{\varepsilon}\right)\left|\nabla\psi_{\varepsilon}\right|dx+\int
J_{\varepsilon}^{2}\,dx.\end{split}$

Using (A.1) to replace the 3nd and 4th integrals on the right-hand side above
yields

$\displaystyle\frac{d}{dt}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]+\varepsilon\int|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dx-\int(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{t}\mathbf{u}_{\varepsilon}\,dx$
(A.6) $\displaystyle=$
$\displaystyle\int(\nabla\cdot\bm{\xi})\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx+\int(\bm{\xi}\cdot\nabla)\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx+\int\nabla\mathbf{H}:(\bm{\xi}-\mathbf{n}_{\varepsilon})\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|dx$
$\displaystyle+\int\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\,dx+\int\nabla\cdot\mathbf{H}\left(\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\right)\,dx$
$\displaystyle+\int\nabla\cdot\mathbf{H}\left(|\nabla\psi_{\varepsilon}|-\bm{\xi}\cdot\nabla\psi_{\varepsilon}\right)\,dx-\int(\nabla\mathbf{H})_{ij}\,\varepsilon\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}\right)\,dx$
$\displaystyle+\int\nabla\mathbf{H}:\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|\,dx+\int
J_{\varepsilon}^{2}\,dx$

We claim that $J_{\varepsilon}^{1}$ arises from the 2nd and 3rd to last
integral. Indeed, when $\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\neq
0$ and $\mathbf{u}_{\varepsilon}\notin B_{\delta_{0}}(\mathfrak{m})$, then

$\displaystyle\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\Pi_{\mathbf{u}_{\varepsilon}}\partial_{j}\mathbf{u}_{\varepsilon}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}\overset{\eqref{projection1},\eqref{ADM
chain
rule}}{=}\partial_{i}\psi_{\varepsilon}\partial_{j}\psi_{\varepsilon}\overset{\eqref{projectionnorm}}{=}n_{\varepsilon}^{i}n_{\varepsilon}^{j}|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2},$

where $(n_{\varepsilon}^{i})_{1\leq i\leq d}=\mathbf{n}_{\varepsilon}$. This
implies

$\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\Pi_{\mathbf{u}_{\varepsilon}}\partial_{j}\mathbf{u}_{\varepsilon}=n_{\varepsilon}^{i}n_{\varepsilon}^{j}|\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}.$
(A.7)

In other two cases defining (2.42), the equation (A.7) also holds. So by
adding zero we find

$\begin{split}&\nabla\mathbf{H}:\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|-(\nabla\mathbf{H})_{ij}\,\varepsilon\left(\partial_{i}\mathbf{u}_{\varepsilon}\cdot\partial_{j}\mathbf{u}_{\varepsilon}\right)\\\
\overset{\eqref{projection1}}{=}&\nabla\mathbf{H}:\mathbf{n}_{\varepsilon}\otimes\mathbf{n}_{\varepsilon}\left|\nabla\psi_{\varepsilon}\right|-\varepsilon(\nabla\mathbf{H})_{ij}(\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon}\cdot\Pi_{\mathbf{u}_{\varepsilon}}\partial_{j}\mathbf{u}_{\varepsilon})\\\
&-(\nabla\mathbf{H})_{ij}\,\varepsilon\Big{(}(\partial_{i}\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\partial_{i}\mathbf{u}_{\varepsilon})\cdot(\partial_{j}\mathbf{u}_{\varepsilon}-\Pi_{\mathbf{u}_{\varepsilon}}\partial_{j}\mathbf{u}_{\varepsilon})\Big{)}\overset{\eqref{key
identity chain},\eqref{J1}}{=}J_{\varepsilon}^{1}\end{split}$

and this finish the proof of the claim.

Using the definition (2.41a) of $\mathbf{n}_{\varepsilon}$, we merge the 2nd
and 3rd integrals on the right-hand side of (A.6) and use
$\nabla\mathbf{H}:(\bm{\xi}\otimes\bm{\xi})$ (due to (2.33)) to obtain

$\displaystyle\frac{d}{dt}E_{\varepsilon}[\mathbf{u}_{\varepsilon}|\Sigma]=-\varepsilon\int|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}\,dx+\int(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{t}\mathbf{u}_{\varepsilon}\,dx$
$\displaystyle+\int(\nabla\cdot\bm{\xi})\,\mathbf{H}\cdot\nabla\psi_{\varepsilon}\,dx+\int\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|\,dx-\int\nabla\mathbf{H}:(\bm{\xi}-\mathbf{n}_{\varepsilon})^{\otimes
2}\left|\nabla\psi_{\varepsilon}\right|\,dx$
$\displaystyle+\int\left(\nabla\cdot\mathbf{H}\right)\Big{(}\frac{\varepsilon}{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{\varepsilon}F(\mathbf{u}_{\varepsilon})-|\nabla\psi_{\varepsilon}|\Big{)}\,dx$
$\displaystyle+\int(\nabla\cdot\mathbf{H})\left(1-\bm{\xi}\cdot\mathbf{n}_{\varepsilon}\right)|\nabla\psi_{\varepsilon}|\,dx+\int(J_{\varepsilon}^{1}+J_{\varepsilon}^{2})\,dx.$
(A.8)

Now we complete squares for the first four terms on the right-hand side of
(A). Reordering terms, we have

$\displaystyle-$
$\displaystyle\varepsilon|\partial_{t}\mathbf{u}_{\varepsilon}|^{2}+(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\partial_{t}\mathbf{u}_{\varepsilon}+(\nabla\cdot\bm{\xi})\mathbf{H}\cdot\nabla\psi_{\varepsilon}+\mathbf{H}_{\varepsilon}\cdot\mathbf{H}|\nabla\mathbf{u}_{\varepsilon}|$
$\displaystyle=-\frac{1}{2\varepsilon}\Big{(}|\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}|^{2}-2(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\cdot\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}+(\nabla\cdot\bm{\xi})^{2}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}\Big{)}$
$\displaystyle\quad-\frac{1}{2\varepsilon}|\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{2\varepsilon}(\nabla\cdot\bm{\xi})^{2}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}+(\nabla\cdot\bm{\xi})\mathbf{H}\cdot\nabla\psi_{\varepsilon}$
$\displaystyle\quad-\frac{1}{2\varepsilon}\Big{(}|\mathbf{H}_{\varepsilon}|^{2}-2\varepsilon|\nabla\mathbf{u}_{\varepsilon}|\mathbf{H}_{\varepsilon}\cdot\mathbf{H}+\varepsilon^{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}|\mathbf{H}|^{2}\Big{)}+\frac{1}{2\varepsilon}\Big{(}|\mathbf{H}_{\varepsilon}|^{2}+\varepsilon^{2}|\nabla\mathbf{u}_{\varepsilon}|^{2}|\mathbf{H}|^{2}\Big{)}$
$\displaystyle=-\frac{1}{2\varepsilon}\Big{|}\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}-(\nabla\cdot\bm{\xi})\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})\Big{|}^{2}-\frac{1}{2\varepsilon}\Big{|}\mathbf{H}_{\varepsilon}-\varepsilon|\nabla\mathbf{u}_{\varepsilon}|\mathbf{H}\Big{|}^{2}-\frac{1}{2\varepsilon}|\varepsilon\partial_{t}\mathbf{u}_{\varepsilon}|^{2}+\frac{1}{2\varepsilon}|\mathbf{H}_{\varepsilon}|^{2}$
$\displaystyle\quad+\frac{1}{2\varepsilon}\Big{(}(\nabla\cdot\bm{\xi})^{2}|\partial\mathrm{d}_{F}(\mathbf{u}_{\varepsilon})|^{2}+2\varepsilon(\nabla\cdot\bm{\xi})\nabla\psi_{\varepsilon}\cdot\mathbf{H}+|\varepsilon\Pi_{\mathbf{u}_{\varepsilon}}\nabla\mathbf{u}_{\varepsilon}|^{2}|\mathbf{H}|^{2}\Big{)}$