section[0pt] *[8pt].

# Regularity from $p$-harmonic potentials to $\infty$-harmonic potentials
in convex rings 00footnotetext:  2020 Mathematics Subject Classification:
35J60, 35J65, 35J70, 49N60 Key words and phases: $p$-harmonic potential,
$\infty$-harmonic potential, convex rings.

Fa Peng, Yi Ru-Ya Zhang and Yuan Zhou

Abstract. The exploration of shape metamorphism, surface reconstruction, and
image interpolation raises fundamental inquiries concerning the $C^{1}$ and
higher-order regularity of $\infty$-harmonic potentials — a specialized
category of $\infty$-harmonic functions. Additionally, it prompts questions
regarding their corresponding approximations using $p$-harmonic potentials. It
is worth noting that establishing $C^{1}$ and higher-order regularity for
$\infty$-harmonic functions remains a central concern within the realm of
$\infty$-Laplace equations and $L^{\infty}$-variational problems.

In this study, we investigate the regularity properties from $p$-harmonic
potentials $u_{p}$ to $\infty$-harmonic potentials $u$ within arbitrary convex
rings $\Omega=\Omega_{0}\backslash\overline{\Omega}_{1}$ in $\mathbb{R}^{n}$.
Here $\Omega_{0}$ is a bounded convex domain in $\mathbb{R}^{n}$ and
$\overline{\Omega}_{1}\subset\Omega_{0}$ is a compact convex set. Our main
results can be summarized as follows:

  * •

We establish interior $C^{1}$ regularity for $\infty$-harmonic potentials,
providing their $C^{1}$-approximation by $p$-harmonic potentials; it answers
an open problem by Lindgren and Lindqvist [43, 44]. We also prove the
existence of streamlines.

  * •

For any real value of $\alpha$, we show that $|Du|^{\alpha}$ belongs to
$W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$. Furthermore, we prove
weak convergence of $D|Du_{p}|^{\alpha}$ to $D|Du|^{\alpha}$ in
$L^{2}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ as $p\to\infty$, along
with $D|Du|^{\alpha}\cdot\frac{Du}{|Du|}=0$ almost everywhere.

  * •

In the degenerate case where $\overline{\Omega}_{1}$ reduces to a single
point, we establish the following equivalence

$\Omega_{0}=B(x_{0},r)$ for some $r>0$ $\Leftrightarrow$ $u\in C^{2}(\Omega)$
$\Leftrightarrow$ $u$ is concave.

  * •

We demonstrate that the distributional second-order derivatives
$\mathcal{D}^{2}u$ are Radon measures with suitable upper bounds. Convergence
of $D^{2}u_{p}$ to $\mathcal{D}^{2}u$ weakly in a measure-theoretic sense is
also established.

These results extend some known findings [39, 43, 44, 56, 58] in two
dimensions. Moreover, in planar convex rings, we prove that $\infty$-harmonic
potentials are twice differentiable almost everywhere, providing optimal
results in this context. The second-order derivatives contribute to the
absolutely continuous part of $\mathcal{D}^{2}u$, enabling

$u(x)=\frac{1}{2}\left(\max_{\overline{B(x,\epsilon)}}u+\min_{\overline{B(x,\epsilon)}}u\right)+o(\epsilon^{2})\quad\mbox{
for almost all $x\in\Omega$ as $\epsilon\to 0$}.$

###### Contents

  1. 1 Introduction
    1. 1.1 Main results: regularity and approximation in convex rings
    2. 1.2 Key ideas to prove main results
    3. 1.3 Equal importance criteria, interpolation by propagation, $\infty$-harmonic potential and approximation by $p$-harmonic potentials
  2. 2 Basic properties of $\infty$-harmonic functions
  3. 3 Quasi-concavity of $p$-harmonic and $\infty$-harmonic potentials
  4. 4 A lower bound of the length of gradients
  5. 5 Proof of Theorem 1.1
  6. 6 Two fundamental inequalities under quasi-concavity
  7. 7 Proof of Theorem 1.3
  8. 8 Proof of Theorem 1.4
  9. 9 Proof of Theorem 1.5
  10. 10 Proof of Theorem 1.6
  11. Appendix A Proof of Corollary 1.2
  12. Appendix B Monotonicity in dimension $n=2$

##  1 Introduction

The $\infty$-Laplace equation, denoted as

$\Delta_{\infty}v:=D^{2}vDv\cdot Dv=0\quad\mbox{in a domain
$U\subset{{{\mathbb{R}}}^{n}}$ with $n\geq 2$,}$

is a nonlinear, highly degenerate second-order elliptic equation,
particularly, one that is not in divergence form. In this context, we will be
working with its viscosity solutions as defined by Crandall-Ishii-Lions in
their work [20]. These solutions are commonly referred to as $\infty$-harmonic
functions in the domain $U$, and you can find the specific definitions in
Section 2 of this text.

The concept of $\infty$-harmonic functions originated with Aronsson in the
1960s [2, 3, 4], as he was investigating the Euler-Lagrange equation for the
absolute minimization of the $L^{\infty}$-functional defined as
$F(v,U)=\sup_{U}|Dv|^{2}$. A function $v\in
W^{1,\infty}_{\mathop{\mathrm{missing}}{\,loc\,}}(U)$ is considered as an
absolute minimizer if the following condition holds:

$F(v,V)\leq F(w,V)\ \mbox{whenever $V\Subset U$ and $w\in
C^{0}(\overline{V})\cap W^{1,\infty}(V)$ with $w|_{\partial V}=v|_{\partial
V}$}.$

The existence of absolute minimizers is discussed in Aronsson’s work [6].
Jensen’s research [36] provided a crucial link between $\infty$-harmonic
functions and absolute minimizers. He also established their uniqueness. For
alternative approaches to prove uniqueness, you can refer to the works of
Barles-Busca [11], Crandall et al [19], Peres et al [53] and Armstrong-Smart
[8].

The $\infty$-Laplace equation is not only a crucial topic in mathematics but
also finds applications in a wide range of fields, including shape
metamorphism, surface reconstruction, image processing, computer vision, tug-
of-war games, Lipschitz learning, and more. For specific references, you can
consult works such as [18, 21, 22, 24, 26, 50, 53].

In shape metamorphism and surface reconstruction, the task of finding suitable
reconstruction functions is of utmost importance and has diverse applications
across scientific disciplines. According to the “equal importance criteria”
introduced by Cong-Parvin [23, 24], these reconstruction functions are
solutions to the Dirichlet problem:

$\Delta_{\infty}u=0$ in $\Omega$; $u=0$ on $\partial\Omega_{0}$ and $u=1$ on
$\partial\overline{\Omega}_{1}$. (1.1)

Here, $\Omega=\Omega_{0}\setminus\overline{\Omega}_{1}$, with $\Omega_{0}$
being a bounded domain in $\mathbb{R}^{n}$, and $\Omega_{1}$ being a subdomain
with its closure $\overline{\Omega}_{1}$ residing within $\Omega_{0}$. In some
cases, $\overline{\Omega}_{1}$ may reduce to a compact connected subset of
$\Omega_{0}$ without an interior. Such a set $\Omega$ is commonly referred to
as a ring domain. Additionally, in the context of the “interpolation algorithm
via propagation” devised by Casas-Torres [26] for image processing,
interpolation functions are also required to solve the Dirichlet problem
(1.1). For more precise details, please refer to Section 1.3 of the relevant
literature.

Jensen’s work [36] provides a significant insight into problem (1.1).
According to Jensen, this problem has a unique viscosity solution, commonly
referred to as the $\infty$-harmonic potential in $\Omega$. Throughout this
paper, we will denote this unique solution as $u$. Notably, $u$ serves as an
$\infty$-harmonic function within the domain $\Omega$ and adheres to the
Dirichlet boundary values $u=0$ on $\partial\overline{\Omega}_{0}$ and $u=1$
on $\partial\overline{\Omega}_{1}$. Since the Dirichlet boundary values are
Lipschitz continuous, it follows that $u$ is in $C^{0,1}(\overline{\Omega})$.

In the context of meeting the regularity requirements for reconstruction
functions in shape metamorphism and surface reconstruction, as well as the
regularity needed for interpolation functions in the interpolation algorithm
via propagation, several questions arise regarding the regularity of
$\infty$-harmonic potentials. These questions, detailed in Section 1.3 for
further motivation, can be summarized as follows:

  * Question 1.1. Can the $\infty$-harmonic potential $u$ in $\Omega$ be shown to have $C^{1}$ regularity, meaning that $u\in C^{1}(\Omega)$?

  * Question 1.2. Does the length of the (partial) derivative of $u$, denoted as $|Du|$, exhibit some Sobolev regularity? Specifically, does $|Du|$ have zero partial derivative along the direction of $Du$ within $\Omega$, as suggested by the equation?

  * Question 1.3. Is it valid to assert that $u$ is twice differentiable almost everywhere? Furthermore, what regularity can be attributed to the distributional second-order derivatives of $u$?

These questions delve into the regularity properties of $\infty$-harmonic
potentials and their derivatives, which are crucial in various applications.
Further insights into their motivations can be found in Section 1.3.

The study of the regularity of $\infty$-harmonic potentials, especially
considering Questions 1.1-1.3, has inherent significance. It not only
addresses important issues but also provides insights into the potential
$C^{1}$ and higher-order regularity of $\infty$-harmonic functions, which is a
central topic in this field. In particular, one of the long-standing
conjectures is as follows:

Conjecture: $\infty$-harmonic functions are continuously differentiable, i.e.,
they are $C^{1}$. (1.2)

Notably, thanks to the $\infty$-harmonic function $x^{4/3}-y^{4/3}$ discovered
by Aronsson [6], it is natural to inquire whether $\infty$-harmonic functions
exhibit $C^{1,1/3}$ regularity and Sobolev $W^{2,{\gamma}}$ regularity with
${\gamma}<3/2$. Several significant contributions in the literature have
addressed these questions, including works by Crandall-Evans [16], Savin [56],
Evans-Savin [30], Evans-Smart [31, 32], Koch wtih two of the authors [39], and
Dong and the authors [27], among others.

Specifically, in two dimensions, Savin [56] proved that $\infty$-harmonic
functions are $C^{1}$, thus confirming the conjecture; see also [60] for a
simpification of the original proof via capacity. Evans-Savin [30]
demonstrated that $\infty$-harmonic functions have $C^{1,{\gamma}}$ regularity
for some $0<{\gamma}<1/3$. Furthermore, in [39] and [27], it was shown that,
for any $\alpha>0$, $|Dv|^{\alpha}$ belongs to $W^{1,2}_{\text{loc}}$,
$D|Dv|^{\alpha}\cdot Dv=0$ almost everywhere, and additionally, $-\det
D[|Dv|^{\alpha}Dv]$ is a nonnegative Radon measure.

In dimensions equal to or greater than three, Evans-Smart [31, 32] established
that $\infty$-harmonic functions are everywhere differentiable. These
contributions collectively advance our understanding of the regularity
properties of $\infty$-harmonic functions and their derivatives.

The established regularity results for $\infty$-harmonic functions naturally
extend to $\infty$-harmonic potentials. Additionally, in the case of two-
dimensional space, when $\overline{\Omega}_{1}$ reduces to a single point
within $\Omega_{0}$, further results have been obtained: By Savin-Wang-Yu [58,
Corollary 1.2], it has been shown that $u$ belongs to $C^{2}(\Omega)$ if and
only if $\Omega$ is of the form $\Omega=B(x_{0},r)$ for some $r>0$. Also in
the work of Lindgren-Lindqvist [43, Theorem 4], it was demonstrated that $u$
does not belong to $C^{1,1}_{\text{loc}}(\Omega)$ if $\Omega_{0}$ is not a
disk. By Brustad [10], the $\infty$-harmonic potential in
$[-1,1]^{2}\setminus\\{0\\}$ is not everywhere twice differentiable. These
findings provide valuable insights into the regularity of $\infty$-harmonic
potentials in specific scenarios.

On the other hand, an interesting aspect of $\infty$-harmonic functions is
their approximation in $C^{0,\gamma}$ and weakly in $W^{1,q}_{\text{loc}}$ by
$p$-harmonic functions that share the same Dirichlet boundary value. This
approximation property has been explored by Bhattacharya-DiBenedetto-Manfredi
[12], as well as in the works of [37, 46].

A function $v\in W^{1,p}(U)$ is termed $p$-harmonic if it serves as a weak
solution to the $p$-Laplace equation:

$\Delta_{p}v:={\rm div}(|Dv|^{p-2}Dv)=0\quad\mbox{in $U$.}$

In the quest for achieving $C^{1}$-regularity or higher-order regularity for
$\infty$-harmonic functions (and, in particular, addressing Conjecture (1.2)),
a natural idea is to establish $C^{1}$-regularity or higher-order regularity
for $p$-harmonic functions uniformly across all large values of $p$. This
would effectively approximate $\infty$-harmonic functions with $p$-harmonic
functions. However, this endeavor presents significant challenges. While
$p$-harmonic functions have been shown to possess $C^{1,\alpha}$ regularity
and some higher-order regularity, these results are heavily dependent on the
specific value of $p$ and are not uniform for all large values of $p$.
(References: [33, 35, 47, 48, 59], among others).

Moreover, in two-dimensional space ($n=2$), although Savin [56] has
demonstrated that $\infty$-harmonic functions are $C^{1}$, the problem of
approximating $\infty$-harmonic functions $v$ in $C^{1}$ by $p$-harmonic
functions $v_{p}$ with the same Dirichlet boundary value, or equivalently,
showing that $v_{p}$ has $C^{1}$-regularity uniformly across all large values
of $p$, remains a challenging open question. The most recent developments, as
presented in [44, 27], aided by [39], have shown that for any $\alpha>0$,
$|Dv_{p}|^{\alpha}$ converges to $|Dv|^{\alpha}$ in $L^{2}_{\text{loc}}$, and
$\det D[|Dv_{p}|^{\alpha}Dv_{p}]$ weakly converges to $\det
D[|Dv|^{\alpha}Dv]$ in the sense of measures. These findings represent
significant progress toward understanding the regularity of $\infty$-harmonic
functions and their approximation by $p$-harmonic functions.

To initiate our exploration, we seek to gain insights into the $C^{1}$ and
higher-order approximation questions for a specific class of $\infty$-harmonic
functions, such as the $\infty$-harmonic potential $u$ within $\Omega$. For
this purpose, let us consider $2<p<\infty$ and introduce $u_{p}$ as a
$p$-harmonic potential within $\Omega$. $u_{p}$ is defined as follows:

$u_{p}$ is a unique $p$-harmonic function in $\Omega$ with Dirichlet boundary
values $u_{p}=i$ on $\partial\Omega_{i}$. In other words, it is the unique
weak solution to the Dirichlet problem:

$\Delta_{p}u=0$ in $\Omega$; $u_{p}=1$ on $\partial\Omega_{1}$ and $u_{p}=0$
on $\partial\Omega_{0}$. (1.3)

Based on the findings of [12], we know that $u_{p}$ converges to $u$ in
$C^{0,\gamma}(\overline{\Omega})$ with $\gamma\in(0,1)$.

In the context of the previous discussions and the motivation provided, we can
now address the following inquiries related to the approximation by
$p$-harmonic functions.

  * Question 1.1A. Is it accurate to assert that $u_{p}$ converges to $u$ in $C^{1}(\Omega)$? Equivalently, does $u_{p}$ belong to $C^{1}(\Omega)$ uniformly for all values of $p$?

  * Question 1.2A. Does $|Du_{p}|$ exhibit Sobolev regularity that is uniform across different values of $p$? Alternatively, does $|Du_{p}|$ converge to $|Du|$ within a certain Sobolev space?

  * Question 1.3A. Can we establish that $u_{p}$ converges to $u$ in specific function spaces with second-order regularity? Moreover, what kind of regularity does the second-order derivative $D^{2}u_{p}$ possess, and is this regularity uniform across a range of values for $p>2$?

These questions aim to explore the regularity properties of $p$-harmonic
functions as they approximate $\infty$-harmonic functions, providing insights
into the convergence and regularity behavior in the context of different
values of $p$.

In the two-dimensional space, i.e. $n=2$, some progress has been made toward
addressing Questions 1.2A and 1.3A as in [44, 27]. Furthermore, in the
specific case where both $\Omega_{0}$ and $\overline{\Omega}_{1}$ are convex,
Lindgren-Lindqvist [43, 44] have made significant strides. They have
demonstrated that $|Du_{p}|$ converges to $|Du|$ locally uniformly. However,
the question of whether $Du_{p}$ converges to $Du$ locally and uniformly
remains an open question, referred to as Question 1A. These developments
represent substantial advancements in our understanding of the regularity and
convergence behavior of $p$-harmonic functions with respect to
$\infty$-harmonic functions in two dimensions.

###  1.1 Main results: regularity and approximation in convex rings

The primary objective of this paper is to offer partial answers to Questions
1.1-1.3 by addressing, in part, Questions 1.1A-1.3A within the context of
arbitrary convex ring domains in dimensions $n\geq 2$.

We say $\Omega$ is a convex ring in ${{{\mathbb{R}}}^{n}}$ with $n\geq 2$ if
$\Omega=\Omega_{0}\setminus\overline{\Omega}_{1}$ is a ring domain, and both
of $\Omega_{0}$ and $\overline{\Omega}_{1}$ are convex. Here
$\overline{\Omega}_{1}$ is a compact subset of $\Omega_{0}$. Denote by $u$ the
$\infty$-harmonic potential in $\Omega$, and for $p\in(2,\infty)$, by $u_{p}$
the $p$-harmonic potential in $\Omega$. We extend $u$ and $u_{p}$ continuously
to whole $\Omega_{0}$ by setting $u=u_{p}=1$ in $\overline{\Omega}_{1}$.

Our initial result is a comprehensive answer to Question 1.1 and Question 1A.

######  Theorem 1.1.

We have $u\in C^{1}(\Omega)$ and $|Du|\neq 0$ in $\Omega$. Moreover $u_{p}\to
u$ in $C^{1}(\Omega)$ as $p\to\infty$.

Consequently, for each $t\in(0,1)$ the level set $\\{x\in\Omega|u(x)=t\\}$ is
a $C^{1}$-regular $(n-1)$-dimensional surface.

In dimension $n=2$, $u\in C^{1}(\Omega)$ can be attributed to the remarkable
result on the $C^{1}$ regularity of $\infty$-harmonic functions established by
Savin [56]. Notably, Theorem 1.1 provides a significant outcome, demonstrating
that $Du_{p}\to Du$ uniformly in each compact subset of $\Omega$. This result
effectively resolves the previously mentioned open question raised by
Lindgren-Lindqvist [43, 44]. It’s worth noting that one should not expect the
uniform convergence of $Du_{p}\to Du$ accross entire domain $\Omega$, as
witted by the example in [44]: In
$\Omega=B(0,1)\setminus\\{0\\}\subset{\mathbb{R}}^{2}$, the $\infty$-harmonic
potential $u(x)=1-|x|$ and the $p$-harmonic potential
$u_{p}(x)=1-|x|^{\frac{p-2}{p-1}}$ with $p>2$. Obviously, the convergence
$Du_{p}\to Du$ is not uniform in $\Omega$.

Theorem 1.1 allows us to establish the following existence of trajectory
(streamline).

######  Corollary 1.2.

Given a point $x\in\Omega$, there is a curve ${\gamma}_{x}\in
C^{0,1}([0,T_{x}];\overline{\Omega})\cap C^{1}([0,T_{x});\Omega)$ for some
$T_{x}\in(0,\infty)$ as a trajectory (streamline) of $u$ starting from $x$ and
terminating at $\partial\Omega_{1}$, that is,

$\displaystyle\frac{d{\gamma}_{x}(t)}{dt}=Du({\gamma}_{x}(t))\quad\forall
t\in[0,T_{x})\quad\mbox{and }\
\quad{\gamma}_{x}(0)=x,\quad{\gamma}_{x}(T_{x})\in\partial\Omega_{1}.$ (1.4)

Moreover, the function $u\circ{\gamma}_{x}$ is convex and its speed
$|Du\circ{\gamma}_{x}|$ is nondecreasing.

In two-dimensional space, Lindgren-Lindqvist’s work, as presented in [43, 44],
not only established the existence of streamlines, as demonstrated in
Corollary 1.2, but also demonstrated their uniqueness. However, in higher
dimensions $n\geq 3$, the question of the uniqueness of ${\gamma}_{x}$ remains
unclear and requires further investigation.

Next, we give partial answers to Question 1.2 and Question 1.2A as below. Here
and below, by abuse of notation, we use $|Du|^{\alpha}$ with $\alpha=0$ to
denote $\ln|Du|$.

######  Theorem 1.3.

For any $\alpha\in{\mathbb{R}}$, one has $|Du|^{\alpha}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$,
$D|Du|^{\alpha}\cdot\frac{Du}{|Du|}=0$ almost everywhere in $\Omega$, and
$|Du_{p}|^{\alpha}\to|Du|^{\alpha}$ weakly in
$W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ as $p\to\infty$.

In dimension $n=2$, Theorem 1.3 is essentially a consequence of [39, 44].
Indeed, when $\alpha>0$, it is a direct consequence of [39, 44]. The case
where $\alpha\leq 0$ then follows from Theorem 1.1 as $|Du|\neq 0$.

Finally, we concentrate on Question 1.3 and Question 1.3A. We begin with the
following equivalence in the degenerate case.

######  Theorem 1.4.

Suppose that $\Omega_{1}=\\{x_{0}\\}$ for some $x_{0}\in\Omega_{0}$. The
following are equivalent:

  1. (i)

$\Omega=B(x_{0},r)\setminus\\{x_{0}\\}$ for some $r>0$.

  2. (ii)

$u\in C^{2}(\Omega)$.

  3. (iii)

$u$ is concave.

In dimension $n=2$, (i)$\Leftrightarrow$(ii) was already proved by Savin-Wang-
Yu [58, Corollary 1.2], where they do not require the convexity of
$\Omega_{0}$. In dimension $n\geq 3$, it is unclear to us whether the
convexity assumption of $\Omega_{0}$ in Theorem 1.4 can be removed.

In a general convex ring domain $\Omega$, we establish the following
regularity for distributional second order derivatives in a measure sense,
which contributes to Question 1.3 and Question 1.3A. The distributional
derivative $\mathcal{D}w=(\mathcal{D}_{i}u)_{i=1}^{n}$ in the sense of
distribution of a function $w\in
L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ is defined via integration
by parts, that is,

$\int_{\Omega}\mathcal{D}_{i}w(x)\phi(x)\,dx=-\int_{\Omega}w(x)\phi_{x_{i}}(x)\,dx\quad\forall\phi\in
C_{c}^{\infty}(\Omega).$

######  Theorem 1.5.

(i) The distributional derivatives $\mathcal{D}^{2}u$ are Radon measures
satisfying

$\int_{\Omega}\langle\mathcal{D}^{2}uDu,Du\rangle\xi\,dx=0\quad\forall\xi\in
C^{\infty}_{c}(\Omega).$

Moreover, $Du\in BV_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$, that is,
the distributional derivatives $\mathcal{D}(Du)$ are Radon measures in
$\Omega$, and $\mathcal{D}(Du)=\mathcal{D}^{2}u$.

(ii) $D^{2}u_{p}\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$
uniformly in $p\in[4,\infty)$ and $D^{2}u_{p}\to\mathcal{D}^{2}u$ weakly in
the sense of measure, and $Du_{p}\to Du$ weakly in
$BV_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$.

Additionally, when $n=2$, we are able to prove the following almost everywhere
twice differentiability of $u$, which also contributes to Question 1.3 and
Question 1.3A.

######  Theorem 1.6.

In dimension $n=2$, $u$ is twice differentiable and $Du$ are differentiable
almost everywhere in $\Omega$. The absolutely continuous part of the measure
$\mathcal{D}^{2}u$ with respect to Lebesgue measure $dx$ is given by
$D^{2}u\,dx$.

Consequently, for almost all $x\in\Omega$ one has $-D^{2}uDu\cdot Du=0$,

$\displaystyle
u(x)=\frac{1}{2}\left[u\left(x+hDu(x)\right)+u\left(x-hDu(x)\right)\right]+o(|h|^{2})\quad\mbox{as
$h\to 0$,}$ (1.5)

and

$\displaystyle
u(x)=\frac{1}{2}\left(\max_{\overline{B(x,\epsilon)}}u+\min_{\overline{B(x,\epsilon)}}u\right)+o(\epsilon^{2})\quad\mbox{
as $\epsilon\to 0$.}$ (1.6)

The above almost everywhere twice differentiability of $u$ is optimal; indeed,
by Brustad [10], the $\infty$-harmonic potential in
$[-1,1]^{2}\setminus\\{0\\}$ is not everywhere twice differentiable.

The concept of an $\infty$-harmonic function in a given domain
$U\subset{\mathbb{R}}^{n}$ is closely related to the satisfaction of equation
(1.6). This equivalence is detailed in the work of Parviainen-Manfredi-Rossi
[52]. However, it’s important to note that, in general, an $\infty$-harmonic
function $v$ does not necessarily satisfy equation (1.6) pointwise. This point
is discussed in [52] as well.

The significance of Theorem 1.6 lies in its assertion that $\infty$-harmonic
potentials within planar convex rings satisfy equation (1.6) almost
everywhere. This result is notable because equation (1.6) was initially
utilized by Oberman [50] for constructing numerical approximations of
$\infty$-harmonic functions. Additionally, it has roots in the Dynamic
Programming Principle as described in [9, 53]. This highlights the practical
utility and theoretical foundations of equation (1.6) in various mathematical
and computational contexts.

###  1.2 Key ideas to prove main results

We are ready to sketch the ideas to our main results. The quasi-concavity of
$u_{p}$(that is convexity of super level sets of $u_{p}$) obtained by Lewis
[41] play a crucial role. Since $u_{p}\to u$ in $C^{0}(\Omega)$, the quasi-
concavity of $u$ follows; see Section 3. In Section 2, we recall several
necessary facts about $\infty$-harmonic functions.

Ideas to prove Theorem 1.1. First, for any $x\in\Omega$, denote by $N_{x}$ the
outer normal unit vectors $\nu$ of the level set $\\{y\in\Omega:u(y)=u(x)\\}$,
that is, $\nu\cdot(y-x)\leq 0$ whenever $u(y)>u(x)$. By choosing some suitable
cones and using comparison with cones from above, we get the following lower
bound

$\displaystyle-Du(x)\cdot\nu\geq\frac{u(x)-u(x+t\nu)}{t}\quad\forall\nu\in
N_{x},\ \forall 0<t<\frac{1}{4}{\rm dist}(x,\partial\Omega_{0}),$ (1.7)

in particular, $|Du(x)|>0$. As a results of this, together with the
differentiability of $u$ at $x$, we deduce that $N_{x}$ consists of the unique
vector $\nu_{x}:=-\frac{Du(x)}{|Du(x)|}$ and hence (1.7) holds with
$\nu=\nu_{x}$; see Lemma 4.1 in Section 4.

Next, thanks to (1.7) with $\nu=\nu_{x}$ and the convexity of
$\Omega_{1}\cup\\{x\in\Omega:u(x)>u(z)\\}$, we are able to obtain the
continuity of $\nu_{x}$ in $x\in\Omega$, that is,

$\nu_{x}\to\nu_{z}\quad\mbox{when $x\to z\in\Omega$. }$

Using this, (1.7) with $\nu_{z}$ and continuity of $u$ we further get

$\displaystyle\liminf_{x\to
z}|Du(x)|\geq\frac{1}{t}[u(z)-u(x+t\nu_{z})]\quad\forall
z\in\Omega,\quad\forall 0<t<\frac{1}{4}{\rm dist}(z,\partial\Omega_{0}),$
(1.8)

Sending $t\to 0$ we have

$\liminf_{x\to z}|Du(x)|\geq|Du(z)|.$

Recalling $\limsup_{x\to z}|Du(x)|\leq|Du(z)|$ by Crandall-Evans-Gariepy [17],
one gets the continuity of $|Du|$ at any $z\in\Omega$. Thanks to
$Du(x)=-|Du(x)|\nu_{x}$, we know $Du\in C^{0}(\Omega)$. See Section 5 for more
details.

Observe that $u_{p}$ also enjoys comparison with cones from above by Lewis
[41]. Since (1.7) holds for $u_{p}$ at any point $x$ and with
$\nu=-\frac{Du_{p}(x)}{|Du_{p}(x)|}$ (see Section 4 for details), by an
argument similar to above, we could show that $Du_{p_{i}}(x_{i})\to Du(x)$
whenever $p_{i}\to\infty$ and $x_{i}\to x$. Thus $Du_{p}\to Du$ locally
uniformly; see Section 5.

Ideas to prove Corollary 1.2. Theorem 1.1 allows us to employ the idea from
Lindgren-Lindqvist [43, 44] in dimension $n=2$ to prove Corollary 1.2. For
reader’s convenience, we give the details in the Appendix A.

Ideas to prove Theorem 1.3. For any smooth quasi-concave function $v$, we
observe a crucial fundamental geometric structural inequality

$\displaystyle 2[|D^{2}vDv|^{2}-\Delta
v\Delta_{\infty}v]\geq|Dv|^{2}[|D^{2}v|^{2}-(\Delta v)^{2}]$ (1.9)

and also a useful divergence formula

$\displaystyle{\rm div}\left(|Dv|^{-2}(\Delta vDv-D^{2}vDv)\right)\geq
0\quad\mbox{when $Dv\neq 0$}.$ (1.10)

Their proofs heavily rely on the convexity of the super level sets of $u$,
which implies the non-positive definite of second fundamental form of its
boundary and the non-positivity of sectional curvatures. See Section 6 for
details.

Applying the divergence formula (1.10) to $u_{p}$ and testing
$|Du_{p}|^{\alpha}\phi^{2}$ for any $\alpha<0$ and any $\phi\in
C^{\infty}_{c}(\Omega)$ we obtain that $|Du_{p}|^{\alpha}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ uniformly in $p>2$. Since
$|Du_{p}|$ is bounded from above and away from zero uniformly in $p$, from
$|Du_{p}|\in W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ uniformly in $p>2$,
we conclude $|Du_{p}|^{\alpha}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ uniformly in $p>2$ for any
$\alpha\in{\mathbb{R}}$. Therefore, together with $C^{1}$-convergence of
$u_{p}\to u$, we conclude $|Du|^{\alpha}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ for all $\alpha$. See Section 7
for details.

Ideas to prove Theorem 1.4. If $\Omega=B(x_{0},r)\setminus\\{x_{0}\\}$, then
the $\infty$-harmonic potential therein is given by $1-\frac{|x-x_{0}|}{r}$,
which is obviously of $C^{2}(\Omega)$ and concave in $B(x_{0},r)$. It then
suffices to show (ii) $\Rightarrow$(i) and (iii) $\Rightarrow$(i). The proof
is given in Section 8. The proof relies on the fact

$\displaystyle\lim_{x\to x_{0}}|Du(x)|=\frac{1}{{\rm
dist}(x_{0},\partial\Omega_{0})},$ (1.11)

which is obtained by using the asymptotic property of $u$ around $x_{0}$ by
Savin-Wang-Yu [58] and Lemma 4.1(that is (1.7) with $\nu=\nu_{x}$).

If $u\in C^{2}$, there exists a unique streamlines ${\gamma}_{x}$ starting
from any point $x\in\Omega$ and ending at $x_{0}$. Since $u\in C^{2}$ implies
$\Delta_{\infty}u=0$ everywhere, we know that $|Du|$ is a constant along
${\gamma}_{x}$, and hence by (1.11), is given by $\frac{1}{{\rm
dist}(x_{0},\partial\Omega_{0})}.$ The speed $|\dot{\gamma}_{x}|$ is also
$\frac{1}{{\rm dist}(x_{0},\partial\Omega_{0})}.$ By a direct calculation one
has $|x-x_{0}|\leq{\rm dist}(x_{0},\partial\Omega_{0})$ for all $x\in\Omega$.
Thus (ii) $\Rightarrow$(i).

If $u$ is concave, then $u$ is twice differentiable almost everywhere. The
concavity of $u$ further show that $D^{2}u(x)\xi\cdot\xi\leq 0$ for all
$\xi\in{{{\mathbb{R}}}^{n}}$ and almost all $x\in\Omega$. This allows us to
get $D^{2}uDu=0$ almost everywhere. Since $D|Du|^{2}=D^{2}uDu$ almost
everywhere and $|Du|^{2}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$, we know that $|Du|$ is
a constant, and hence by (1.11), is given by $\frac{1}{{\rm
dist}(x_{0},\partial\Omega_{0})}.$ With this in hand, via streamline we do
prove that $\Omega=B(x_{0},r)$ similarly, that is, (iii) $\Rightarrow$(i).

Ideas to prove Theorem 1.5. Applying (1.9) to $u_{p}$ and using the equation
$-\Delta_{p}u=0$, we derive that

$|D^{2}u_{p}|\leq 2|\Delta u_{p}|+2|D|Du_{p}||\quad\forall p\in[4,\infty).$

Since $-\Delta u_{p}$ is nonnegative and $|D|Du_{p}||\in
L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ uniformly in
$p\in[4,\infty)$, we know that $|D^{2}u_{p}|\in
L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ uniformly in
$p\in[4,\infty)$. So $D^{2}u_{p}$ weakly converges to some Radon measure
$\mu$. Note that $\mathcal{D}^{2}u$ coincides with $\mu$ in the sense of
distribution. Moreover, we also derived a quantitative upper bound

$\displaystyle\|\mathcal{D}^{2}u\|(B(x,r))\leq 2[-\Delta
u](B(x,2r))+2\int_{B(x,2r)}|D|Du||\,dx,$ (1.12)

where $-\Delta u$ is a nonnegative Radon measure. Here the measure $\|\mu\|$
stands for the total variation of a signed Radon measure $\mu$. If $\mu$ is
nonnegative, we write $\|\mu\|=\mu$.

Ideas to prove Theorem 1.6. To obtain almost everywhere twice
differentiability of $u$, it suffices to prove almost everywhere
differentiability of $Du$. By Rademacher’s theorem [54] (see also [55]), one
only needs to show the pointwise Lipschitz constant
${\mathop{\mathrm{missing}}{\,Lip}}(Du)(x)<\infty$ for almost all
$x\in\Omega$.

To this end, we observe a monotonicity property of $Du$ in Appendix B, which
allows us to bound

${\mathop{\mathrm{missing}}{\,Lip}}(Du)(x)\leq\limsup_{r\to
0}\frac{\|\mathcal{D}^{2}u\|(B(x,r))}{r^{2}}.$

Considering (1.12), noting $|D|Du||\in
L^{2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ implies

$\lim_{r\to 0}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to25.56952pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to25.56952pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to18.7401pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to18.7401pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(x,2r)}|D|Du||\,dx<\infty\
\mbox{for almost all $x\in\Omega$,}$

we still need to check

$\limsup_{r\to 0}\frac{[-\Delta u](B(x,r))}{r^{2}}<\infty\ \mbox{for almost
all $x\in\Omega$}.$

This holds because the singular part $[-\Delta u]_{s}$ of the measure $-\Delta
u$ satisfies

$\frac{[-\Delta u]_{s}(B(x,2r))}{r^{2}}\to 0\ \mbox{for almost all
$x\in\Omega$}.$

###  1.3 Equal importance criteria, interpolation by propagation,
$\infty$-harmonic potential and approximation by $p$-harmonic potentials

The need to reconstruct intermediate shapes that gradually transition from a
given source shape to a desired target shape is a fundamental requirement in a
variety of scientific fields. This process forms a critical component in the
study of shape metamorphism, surface reconstruction, image interpolation, and
other related areas. Relevant references include [18, 21, 22, 23, 24, 51, 57],
among others.

To be more specific, consider two distinct compact $(n-1)$-dimensional
surfaces, denoted as $S_{0}$ and $S_{1}$. The objective is to construct a
family $\\{S(t)\\}_{s\in[0,1]}$ of $(n-1)$-dimensional surfaces so that
$S(0)=S_{0}$, $S(1)=S_{1}$, and $S(t)$ maintains continuity for $t\in[0,1]$.
Additionally, for each $t\in(0,1)$, $S(t)$ is expected to exhibit a certain
level of smoothness, and the entire family $\\{S_{t}\\}_{t\in[0,1]}$ is
intended to span the metamorphism region
$\overline{\Omega}=(\overline{\Omega}_{0}\setminus\Omega_{1})\cup(\overline{\Omega}_{1}\setminus\Omega_{0})$,
where $\Omega_{i}$ is the domain enclosed by $S_{i}$ for $i=0,1$. This process
is crucial for a wide range of applications, facilitating the smooth
transition between different shapes while ensuring continuity and smoothness
in the intermediate shapes.

The goal is to represent the desired intermediate surfaces using a
reconstruction and interpolation function $u:\overline{\Omega}\to{\mathbb{R}}$
satisfying $\\{x\in\overline{\Omega}|u(x)=i\\}=S_{i}$ for $i=0,1$ and hence
the Dirichlet boundary condition

$u=1$ on $S_{1}=\partial\Omega_{1}$ and $u=0$ on $S_{0}=\partial\Omega_{0}$,
(1.13)

where it is assumed that,

$S_{0}\cap S_{1}=\emptyset$ and $\Omega\neq\emptyset$, that is, either
$\overline{\Omega}_{0}\subset\Omega_{1}$ or
$\overline{\Omega}_{1}\subset\Omega_{0}$. (1.14)

Without loss of generality, we can assume that
$\overline{\Omega}_{1}\subset\Omega_{0}$.

For any reconstruction or interpolation function $u$ that meets the criteria
of $u\in C^{1}(\Omega)\cap C^{0}(\overline{\Omega})$ and $|Du|\neq 0$ in
$\Omega$, the implicit function theorem allows us to construct a collection of
level surfaces:

$\left\\{\\{x\in\overline{\Omega}|u(x)=t\\}\right\\}_{t\in[0,1]}.$

These level surfaces provide a means to select intermediate surfaces or
interpolation surfaces between the images $S_{0},S_{1}$. One may also view

$\mathcal{S}=\\{(x,z)\in\overline{\Omega}\times[0,1]|u(x)=z,z\in[0,1]\\}=\cup_{t\in[0,1]}S(t)\times\\{t\\}$

as a reconstructed $n$-dimensional surface from cross-sectional
$(n-1)$-dimensional surfaces $S_{0}$ and $S_{1}$, that is,
$\mathcal{S}\cap{\mathbb{R}}^{n}\times\\{i\\}=S_{i}$ for $i=0,1$. This
approach allows for the smooth construction of intermediate surfaces based on
the given boundary conditions and the choice of the interpolation function.

The generation of suitable reconstruction functions and the creation of
intermediate shapes that fulfill various requirements are fundamental
challenges in diverse scientific fields. The literature, as found in
references such as [22, 21, 24, 50], has introduced several critical
constraint principles for the reconstruction process. Correspondingly, various
numerical methods have been developed and widely applied.

It’s important to note that different sets of constraints can lead to the
study of solutions to different partial differential equations. This
demonstrates the versatility and adaptability of reconstruction techniques to
address the specific needs of different scientific domains.

In the absence of any information about the reconstruction process, we expect
that every point in $\Omega$ is equally important and contributes similarly to
the reconstruction process, and that any other assumption means that we know
something about reconstruction functions. This natural constraint is now known
as equal importance criteria, which was originally introduced by [18, 22, 24]
in the study of shape metamorphism and surface reconstruction. It was further
formulated therein by requiring that the change of the reconstruction function
$u$ in the gradient-magnitude along direction is $0$, that is,

$\displaystyle D|Du|\cdot\frac{Du}{|Du|}=0\quad\mbox{in $\Omega$}.$ (1.15)

This implies, along each trajectory of the gradient of $u$, the magnitude of
the gradient is a constant; in another words, the height of $u$ decrease
linearly from 1 to 0, and the level surfaces of $u$ are equally distributed
along the gradient. Together with Dirichlet boundary condition, the equal
importance criteria leads to the problem:

Find a function $u:\overline{\Omega}\to{\mathbb{R}}$ to solve (1.15) with
Dirichlet boundary condition (1.13). (1.16)

We expect that a solution $u$ to (1.16) should enjoy a nice regularity that
$u\in C^{0}(\overline{\Omega})\cap C^{1}(\Omega)$, $Du\neq 0$ and $|Du|$ has
partial derivative $0$ along the direction $Du$.

Moreover, to find a suitable interpolation function as required by image
process, a remarkable interpolation algorithm via propagation was introduced
by Casas and Torres [26]; see also Caselles-Morel-Sbert [22]. Suppose that $u$
is the corresponding interpolation function obtained by this algorithm. If $u$
is $C^{2}$ (even twice differentiable) at $x$, the propagation here requires
that

$\displaystyle u(x)=\frac{1}{2}\left[u(x+hDu(x))-u(x-hDu(x))\right]+o(h^{2}).$
(1.17)

By Taylor expansion, letting $h\to 0$ one gets

$\displaystyle\Delta_{\infty}u(x):=D^{2}u(x)Du(x)\cdot Du(x)=0.$ (1.18)

This enables us to consider the problem:

Find a function $u:\overline{\Omega}\to{\mathbb{R}}$ to solve (1.18) in
$\Omega$ with Dirichlet boundary condition (1.13). (1.19)

Note the above algorithm does not yield a $C^{2}$ interpolation function
necessarily, even which is generally desirable in numerical analysis. One may
ask whether interpolation functions or solutions to (1.19) are twice
differentiable almost everywhere so that (1.17) and (1.18) hold almost
everywhere.

$x$$x+hDu(x)$$x-hDu(x)$${\rm FIG\ 1:Interpolation\ by\ propagation}$

The equation $\Delta_{\infty}u=0$ given in (1.18) is exactly the
$\infty$-Laplacian equation as given in Section 1.1, and its viscosity
solutions are called $\infty$-harmonic functions. The equation (1.15) is
written as the the normalized $\infty$-Laplacian equation
$\Delta^{N}_{\infty}u=0$, where

$\Delta^{N}_{\infty}u:=\frac{\Delta_{\infty}u}{|Du|^{2}}=D|Du|\cdot\frac{Du}{|Du|}\quad\mbox{when
$u\in C^{2}$ and $Du\neq 0$}.$

By Peres-Schramm-Sheffield-Wilson [53], $\infty$-harmonic functions are
exactly viscosity solutions to the equation $\Delta^{N}_{\infty}u=0$. For more
background of $\infty$-harmonic functions we refer to the beginning of Section
1. Therefore, the problem (1.16) and the problem (1.19) lead to the Dirichlet
problem

$\Delta_{\infty}u=0$ in $\Omega$; $u=1$ on $S_{1}=\partial\Omega_{1}$ and
$u=0$ on $S_{0}=\partial\Omega_{0}$.

By Jensen [36], the $\infty$-harmonic potential in $\Omega$ is the unique
viscosity solution to such Dirichlet problem, and therefore provides a
viscosity solution to the problem (1.19), and also a “ weak” solution to
(1.16). The regularity required by reconstruction functions in the problem
(1.16) and by interpolation functions in the problem (1.19) lead to the study
of the regularity of $\infty$-harmonic potential, and in particular, Questions
1.1-1.3. Our finding in Section 1.1 gives a limited insightful understanding
of theses questions.

Since Jensen [36] identified $\infty$-harmonic functions with absolute
minimizers for $L^{\infty}$-functional
$F(v,\Omega)=\||Dv|^{2}\|_{L^{\infty}(\Omega)}$. The equal importance criteria
somehow ask that the reconstruction functions is absolute minimizer for
$L^{\infty}$-functional $F(v,\Omega)$ by Aronsson in 1960’s [2, 3, 4].
Meanwhile, in the shape metamorphism and surface reconstruction, it is also
quite often to get the reconstruction via the constraint minimizing the
$L^{p}$-functional

$E_{p}(v,\Omega)=\frac{1}{p}\int_{\Omega}|Dv|^{p}\,dx,$

where $1<p<\infty$; see Cong-Esser-Parvin-Bebis [18]. The corresponding Euler-
Lagrange equation reads as $\Delta_{p}v=0$ in $\Omega$, where the
$p$-Laplacian

$\Delta_{p}v={\rm div}(|Dv|^{p-2}Dv).$

This leads to the Dirichlet problem

$\displaystyle\Delta_{p}v=0\mbox{ in $\Omega$};\ v=0\ {\rm on}\
\partial\Omega_{0},\ v=1\ {\rm on}\ \partial\Omega_{1}.$ (1.20)

This problem admits a unique weak solution $u_{p}\in W^{1,p}(\Omega)\cap
C^{0}(\overline{\Omega})$. This also arises from the an axiomatic approach to
image interpolation, in Caselles-Morel-Sbert[22]. Such $u_{p}$ was already
known to have some better regularity, say $C^{1,{\gamma}}$-regularity and also
some higher order regularity but all of them are not uniformly in all
$p\in(2,\infty)$.

It is valuable to comprehend both the similarities and distinctions among
reconstruction functions obtained by different approaches. Observe that,
letting $p\to\infty$, the energy functional $E_{p}(v,\Omega)$ goes to the
$L^{\infty}$-functional $F(v,\Omega)$; the normalized $p$-Laplacian
$\frac{1}{p}|Dv|^{2-p}\Delta_{p}v$ converges to the normalized
$\infty$-Laplacian $\Delta_{\infty}^{N}v$ formally; the $p$-harmonic
potentials $u_{p}$ converges to the $\infty$-harmonic potentials $u$ in
$C^{0,{\gamma}}(\Omega)$ and weakly in $W^{1,q}(\Omega)$. However, to consider
the regularity and stability required by numeric analysis, some higher order
approximation are expected. Our answers to Question 1.1A-1.3A partially solve
these problems.

##  2 Basic properties of $\infty$-harmonic functions

In this section we recall the definition of $\infty$-harmonic functions and
also several useful properties. Let $U$ be an arbitrary bounded domain of
${{{\mathbb{R}}}^{n}}$.

######  Definition 2.1.

A function $v\in C^{0}(U)$ is a viscosity subsolution to

$\displaystyle-\Delta_{\infty}w=0\quad{\rm in}\quad U$ (2.1)

if at any $x_{0}\in U$, for any $\phi\in C^{2}(U)$ satisfying

$v(x_{0})-\phi(x_{0})\geq v(x)-\phi(x)\quad\forall x\in B(x_{0},r)\Subset U\
\mbox{for some $r>0$},$

one has $-\Delta_{\infty}\phi(x_{0})\leq 0.$

A function $v\in C^{0}(U)$ is a viscosity supersolution to (2.1) if $-v$ is a
viscosity supersolution of (2.1).

A function $v\in C^{0}(U)$ is a viscosity solution to (2.1) if it is both a
viscosity subsolution and a viscosity supersolution to (2.1).

Viscosity solutions to (2.1) are called $\infty$-harmonic functions in $U$.

The following property is well-known; see for example Crandall [15].

######  Theorem 2.2.

If $v\in C^{0,1}(\overline{U})$ is $\infty$-harmonic in $U$, then
${\mathop{\mathrm{missing}}{\,Lip}}(v,U)={\mathop{\mathrm{missing}}{\,Lip}}(v,\partial
U)$.

Jensen [36] established a comparison principle.

######  Theorem 2.3.

If $v_{1}\in C^{0,1}(\overline{U})$ is viscosity subsolution to (2.1) in $U$
and $v_{2}\in C^{0,1}(\overline{U})$ is viscosity supersolution to (2.1) in
$U$, then

$\max_{\overline{U}}[v_{1}-v_{2}]=\max_{\partial U}[v_{1}-v_{2}].$

The comparison with cones was introduced by Crandall-Evans-Gariepy [17].

######  Definition 2.4.

A function $v\in C^{0}(U)$ enjoys comparison with cones from above in $U$ if
for any $V\Subset U$, $x_{0}\in U$, and $b\geq 0$, it holds that

$v(x)\leq v(x_{0})+b|x-x_{0}|\quad\forall x\in V$ whenever $v(x)\leq
v(x_{0})+b|x-x_{0}|\quad\forall x\in\partial(V\backslash\\{x_{0}\\}).$

A function $v\in C^{0}(U)$ enjoys comparison with cones from below in $U$ if
$-v$ enjoys comparison with cones from above in $U$.

A function $v\in C^{0}(U)$ enjoys comparison with cones in $U$ if it enjoys
both comparison with cones from below and from above.

A crucial fact for $\infty$-harmonic functions is that they can be
characterised through comparison with cones; see Crandall-Evans-Gariepy [17].

######  Theorem 2.5.

Let $v\in C^{0}(U)$. Then $v$ is a subsolution to (2.1) in $U$ if and only if
$v$ satisfies comparison with cones from above in $U$.

Consequently, $v$ is $\infty$-harmonic if and only if $v$ satisfies comparison
with cones.

For $x\in U$ and $0<r<{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial U)$,
define the slope functions

$S^{+}_{r}(v,x):=\max_{y\in\partial(B(x,r)\cap
U)\backslash\\{x\\}}\frac{v(y)-v(x)}{r}$

and

$S^{-}_{r}(v,x):=\max_{y\in\partial(B(x,r)\cap
U)\backslash\\{x\\}}\frac{v(x)-v(y)}{r}.$

Denote by ${\mathop{\mathrm{missing}}{\,Lip}}\,v(x)$ the pointwise Lipshictz
constant at $x$, that is,

${\mathop{\mathrm{missing}}{\,Lip}}\,v(x):=\limsup_{r\to 0}\sup_{|x-y|\leq
r}\frac{|u(y)-u(x)|}{r}.$

Crandall-Evans-Gariepy [17] obtained the monotonicity of the slope functions.

######  Lemma 2.6.

Let $v$ be upper semi-continuous in $U$ and enjoy comparison with cones from
above. Then for any $x\in U$, $S^{+}_{r}(v,x)$ is nondecreasing in
$r\in(0,{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial U))$. Moreover,

$\lim_{r\to
0}S^{+}_{r}(v,x)={\mathop{\mathrm{missing}}{\,Lip}}\,v(x)\quad\mbox{for all
$x\in U$.}$

As a consequence of (2.5), the following strong maximum principle can be found
in Crandall [15, Section 4].

######  Corollary 2.7.

Let $v$ be an $\infty$-harmonic function in a domain $U$. If
$v(x)=\max_{\overline{B(x,r)}}v$ for some ball $B(x,r)\Subset U$, then $v$ is
a constant on $B(x,r/2)$.

Consequently, if $v$ attains its maximum at any interior point of $U$, then it
must be a constant.

Note that $p$-harmonic functions satisfies a similar strong maximum principle;
see Lindqvist [42].

Evans-Smart [31, 32] proved the everywhere differentiability of
$\infty$-harmonic functions.

######  Theorem 2.8.

If $v$ is an $\infty$-harmonic function in a domain $U$, then $v$ is
everywhere differentiable in $U$. Consequently,
${\mathop{\mathrm{missing}}{\,Lip}}\,v(x)=|Dv(x)|$ for all $x\in U$.

##  3 Quasi-concavity of $p$-harmonic and $\infty$-harmonic potentials

In this paper, unless other specify, we always assume that $\Omega$ is a
convex ring, that is, $\Omega=\Omega_{0}\setminus\overline{\Omega}_{1}$,
$\Omega_{0}$ is bounded convex domain and $\overline{\Omega}_{1}$ is a convex
compact subset of $\Omega_{0}$. We always denote by $u\in
C^{0}(\overline{\Omega})\cap W^{1,\infty}(\Omega)$ the $\infty$-harmonic
potential in $\Omega$. Naturally, one extends $u$ to the whole domain
${{{\mathbb{R}}}^{n}}$ by setting

$u=1$ in $\overline{\Omega}_{1}$ and $u=0$ in
${{{\mathbb{R}}}^{n}}\setminus\Omega_{0}$.

For each $p\in(2,\infty)$, we always denote by $u_{p}\in
C^{0}(\overline{\Omega})\cap W^{1,p}(\Omega)$ the $p$-harmonic potential in
$\Omega$, that is, the unique weak solution to (1.20). Naturally, one extends
$u_{p}$ to the whole domain ${{{\mathbb{R}}}^{n}}$ by setting

$\mbox{$u_{p}=1$ in $\overline{\Omega}_{1}$ and $u_{p}=0$ in
${{{\mathbb{R}}}^{n}}\setminus\Omega_{0}$}.$

Let us review some result of $u_{p}$ due to Lewis [41, Theorem 1]. A function
$f\in C^{0}({{{\mathbb{R}}}^{n}})$ is called quasi-concave in
${{{\mathbb{R}}}^{n}}$ if for each $t\in{\mathbb{R}}$ the super level set
$\\{x\in{{{\mathbb{R}}}^{n}}:f(x)>t\\}$ is convex whenever it is not empty
set, or equivalently,

$f(\lambda
x+(1-\lambda)y)\geq\min\\{f(x),f(y)\\}\quad\forall\lambda\in[0,1],x,y\in{{{\mathbb{R}}}^{n}}.$

######  Lemma 3.1.

Let $p\in[2,\infty)$. It holds that $u_{p}\in C^{\infty}(\Omega)$ with
$Du_{p}\neq 0$ in $\Omega$, $u_{p}$ is quasi-concave in
${{{\mathbb{R}}}^{n}}$, and $-\Delta u_{p}>0$ in $\Omega$.

Consequently, we have the following.

######  Lemma 3.2.

Let $p\in[2,\infty)$. It holds that $u_{p}$ is a viscosity subsolution to
(2.1) in $\Omega$, and hence $u_{p}$ enjoys the comparison with cones from
above in $\Omega$. Moreover, $u_{p}\leq u$ in $\Omega$.

###### Proof.

Using $-\Delta_{p}u_{p}=0$ and $\Delta u_{p}<0$ in $\Omega$ for all $p>2$, we
have

$-\Delta_{\infty}u_{p}=\frac{1}{p-2}\Delta u_{p}|Du_{p}|^{2}\leq 0\quad{\rm
in}\quad\Omega.$

This then guarantees that $u_{p}$ enjoys the comparison with cones from above.
By the comparison principle, we have $u_{p}\leq u$ in $\Omega.$ ∎

Observe that $0<u_{p}<1$ in $\Omega$ via the strong maximum principle, and
therefore,

$\Omega_{0}=\\{x\in\overline{\Omega}_{0}|u_{p}(x)>0\\}.$

For each $t\in[0,1)$, we write the super level set

$\Omega_{t}^{p}:=\\{x\in\Omega_{0}|u_{p}(x)>t\\}$

and the level set

$S^{p}_{t}:=\\{x\in\Omega_{0}|u_{p}(x)=t\\}.$

Note that $\Omega^{p}_{0}=\Omega_{0}$. As a consequence of Lemma 3.2, one has
the following.

######  Lemma 3.3.

For each $t\in[0,1)$, $\Omega_{t}^{p}$ is convex, and its boundary
$\partial\Omega_{t}^{p}=S^{p}_{t}$. If $t\in(0,1)$, then
$S^{p}_{t}\subset\Omega$.

The following was well-known; see [12], [46] and [37, Lemma 3.2].

######  Lemma 3.4.

As $p\to\infty$, one has $u_{p}\to u$ in $C^{0,{\gamma}}(\overline{\Omega})$
and $Du_{p}\to Du$ weakly in $L^{q}(\Omega)$ for any ${\gamma}\in(0,1)$ and
$q\geq 1$.

Moreover, it follows by the strong maximum principle that $0<u(x)<1$ for
$x\in\Omega$, and hence

$\Omega_{0}=\\{x\in\Omega_{0}|u(x)>0\\}$ and
$\partial\Omega_{0}=\\{x\in\overline{\Omega}_{0}|u(x)=0\\}$.

For each $t\in[0,1)$, we write the super level set

$\Omega_{t}:=\\{x\in\Omega_{0}:u(x)>t\\},$

and the level set

$S_{t}:=\\{x\in\Omega_{0}:u(x)=t\\}.$

Via the fact that $u_{p}\to u$ in $C^{0}(\overline{\Omega})$ as $p\to\infty$
and the strong maximum principle, we obtain the following property; see also
[43, 44]. For reader’s convenience, we give details of its proof.

######  Lemma 3.5.

For each $t\in[0,1)$, $\Omega_{t}$ is convex, and moreover, its boundary
$\partial\Omega_{t}=S_{t}$. If $t\in(0,1)$, then $S_{t}\subset\Omega_{0}$.

###### Proof.

Let $t\in(0,1)$. By the continuity of $u$, $\Omega_{t}$ is a non-empty domain
(connected open subset of $\Omega_{0}$). Since $u_{p}\to u$ in
$C^{0}(\overline{\Omega}_{0})$, the convexity of the super level set of
$u_{p}$ yields the convexity of $\Omega_{t}$. By $u\in
C^{0}(\overline{\Omega}_{0})$ and $0<u<1$ in $\Omega$, we know that

$\partial\Omega_{t}\subset S_{t}$ and
$\Omega_{t}\subset\overline{\Omega}_{t}\subset\Omega_{s}$ whenever $0\leq
s<t<1$.

Below we show that $S_{t}\subset\partial\Omega_{t}$ by contradiction. Assume
that $S_{t}\setminus\partial\Omega_{t}\neq\emptyset$. Write
$E_{t}:=\\{x\in\Omega_{0},u(x)\geq t\\}$. Noting that
$\overline{\Omega}_{t}=\Omega_{t}\cup\partial\Omega_{t}$, we have

$E_{t}\setminus\overline{\Omega}_{t}=S_{t}\setminus\partial\Omega_{t}\neq\emptyset.$

Moreover, the convexity of $\Omega_{t}$ yields the convexity of
$\overline{\Omega}_{t}$. Similarly, the convexity of $\Omega_{t}$ and
continuity of $u$ further lead to the convexity of $E_{t}$. Thus
$(E_{t})^{\circ}\setminus\overline{\Omega}_{t}\neq\emptyset$. Indeed, take any
$x_{0}\in E_{t}\setminus\overline{\Omega}_{t}$. By the convexity of
$\overline{\Omega}_{t}$, there is unique $y_{0}\in\overline{\Omega}_{t}$ such
that $|x_{0}-y_{0}|={\rm dist}(x_{0},\overline{\Omega}_{t})>0$. Using a
continuity of distance function ${\rm dist}(\cdot,\partial\Omega_{0})$, we can
find $\lambda_{t}\in(0,1)$ such that for any $\lambda\in(\lambda_{t},1)$,

${\rm dist}(\lambda x_{0}+(1-\lambda)z,\overline{\Omega}_{t})\geq{\rm
dist}(x_{0},\overline{\Omega}_{t})-|\lambda
x_{0}+(1-\lambda)z-x_{0}|>0,\quad\forall z\in\overline{\Omega}_{t}.$

That is, the open set

$\\{\lambda
x_{0}+(1-\lambda)z|\lambda\in(\lambda_{t},1),z\in\overline{\Omega}_{t}\\}\subset
E_{t}\setminus\overline{\Omega}_{t}.$

Therefore in the domain $(E_{t})^{\circ}\setminus\overline{\Omega_{1}}$, $u$
reaches its infimum$t$ at some interior point. However, since $u$ is
$\infty$-harmonic in $(E_{t})^{\circ}\setminus\overline{\Omega_{1}}$, by the
strong maximum principle and a covering argument, we deduced that $u\equiv t$
on $\Omega$ with $0<t<1$. This leads to a contradiction. ∎

##  4 A lower bound of the length of gradients

For an $\infty$-harmonic potential $u$ in a convex ring $\Omega$, given any
$x\in\Omega$, denote by $N_{x}$ the collection of all unit vector
$\nu\in{\mathbb{R}}^{n}$ so that

$\nu\cdot(y-x)\leq 0\quad\forall y\in\Omega_{u(x)}.$

Since $\Omega_{u(x)}$ is convex, one has $N_{x}\neq\emptyset$. In other words,
$N_{x}$ is the collection of outer normal directions $\nu$ of all supporting
hyperplane of $\Omega_{u(x)}$ at $x$, that is, $\Omega_{u(x)}$ lies in the
side $\\{y\in{{{\mathbb{R}}}^{n}}|(y-x)\cdot\nu<0\\}$ of the hyperplane
$P_{\nu}=\\{y\in{{{\mathbb{R}}}^{n}}|(y-x)\cdot\nu=0\\}$.

We have the following lower bound of the partial derivative of $u$ along any
direction in $N_{x}$. Note that, given any unit vector $\nu$, it follows from
the convexity of $\Omega_{0}$ that there is an unique $r_{x,\nu}\in(0,\infty)$
such that

$x+r_{x,\nu}\nu\in\partial\Omega_{0}$, that is, $u(x+r_{x,\nu}\nu)=0$. (4.1)

If $\nu\in N_{x}$, one then has

$\displaystyle{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial\Omega_{0})\leq
r_{x,\nu}\leq{\mathop{\mathrm{missing}}{\,diam\,}}(\Omega_{0}).$ (4.2)

######  Lemma 4.1.

For any $x\in\Omega$, it holds

$\displaystyle-
Du(x)\cdot\nu\geq\frac{u(x)-u(x+r\nu_{x})}{r}>0\quad\forall\nu\in N_{x},\
r\in(0,r_{x,\nu}].$ (4.3)

Moreover,

$\displaystyle N_{x}=\left\\{\nu_{x}:=-\frac{Du(x)}{|Du(x)|}\right\\}.$ (4.4)

Consequently,

$\displaystyle|Du(x)|\geq\frac{u(x)-u\left(x+r\nu_{x}\right)}{r}\quad\forall
r\in(0,r_{x,\nu_{x}}].$ (4.5)

In particular,

$\displaystyle|Du(x)|\geq\frac{u(x)}{r_{x,\nu_{x}}}.$ (4.6)

###### Proof.

Proof of (4.3). Assume that $x\in\Omega$. Given any $r\in(0,r_{x,\nu}]$, as
mentioned above we have
$z=x+r\nu\in\overline{\Omega}_{0}\setminus\Omega_{u(x)}$. Then it follows from
the convexity of $\Omega_{u(x)}$ that
$\overline{B(z,r)}\cap\overline{\Omega_{u(x)}}=\\{x\\}$ and
$B(z,r)\cap\Omega_{u(x)}=\emptyset$. Therefore

$\displaystyle\max_{\overline{B(z,r)\cap\Omega_{0}}}u=u(x),$

and then

$u(y)\leq u(z)+u(x)-u(z)=u(z)+\frac{u(x)-u(z)}{r}|y-z|\quad\forall
y\in\partial B(z,r)\cap\overline{\Omega}_{0}.$

Since $u(x)\geq u(z)$, we get

$\displaystyle u(y)=0\leq u(z)+\frac{u(x)-u(z)}{r}|y-z|,\quad\forall
y\in\partial\Omega_{0}.$

Noting $\partial(B(z,r)\cap\Omega_{0})\subset(\partial
B(z,r)\setminus\overline{\Omega}_{0})\cup\partial\Omega_{0}$, and by applying
comparison with cones from above we obtain

$\displaystyle u(y)\leq u(z)+\frac{u(x)-u(z)}{r}|y-z|,\quad\forall y\in
B(z,r)\cap\Omega_{0}.$

Recalling $z=x+r\nu$, then by setting $y=x+t\nu$ for $t\in(0,r)$ one has

$\displaystyle u(x+t\nu)$ $\displaystyle\leq
u(z)+\frac{u(x)-u(z)}{r}|x+t\nu-z|$
$\displaystyle=u(z)+(r-t)\frac{u(x)-u(z)}{r}=u(x)-t\frac{u(x)-u(z)}{r},$

that is,

$\frac{u(x)-u(x+t\nu)}{t}\geq\frac{u(x)-u(x+r\nu)}{r}.$

By letting $t\to 0$, it holds that

$|Du(x)|\geq-Du(x)\cdot\nu\geq\frac{u(x)-u(x+r\nu)}{r}>0,$

that is, (4.3) holds.

Proof of (4.4). In order to prove $\nu_{x}\in N_{x}$, it suffices to show that
$Du(x)\cdot(y-x)\geq 0$ for all $y\in\Omega_{u(x)}$. For any
$y\in\Omega_{u(x)}$, via the differentiability of $u$ in $\Omega$ we have that

$Du(x)\cdot\frac{y-x}{|y-x|}=\lim_{t\to
0^{+}}\frac{u\left(x+t\frac{y-x}{|y-x|}\right)-u(x)}{t}.$

It follows by the convexity of $\Omega_{u(x)}$ that

$x+t\frac{y-x}{|y-x|}=(1-\frac{t}{|y-x|})x+\frac{t}{|y-x|}y\in\Omega_{u(x)}\quad\forall
0<t<|y-x|.$

Thus

$u(x+t\frac{y-x}{|y-x|})>u(x)\quad\mbox{and then $Du(x)\cdot(y-x)\geq 0$.}$

Next, we show that $N_{x}=\\{\nu_{x}\\}$ by contradiction. Assume that there
exists another $\nu\in N_{x}$ with $\nu\neq\nu_{x}$. We claim that there is a
unit vector $\eta$ such that $\eta\cdot\nu_{x}<0$ while $\eta\cdot\nu>0$.
Indeed, $\nu$ and $\nu_{x}$ span a 2-dimension plane $P$. If
$\nu\cdot\nu_{x}<0$ we take $\eta=\nu$. If $\nu\cdot\nu_{x}\geq 0$, denote by
$\theta$ the angle between $\nu$ and $\nu_{x}$, we let $\eta$ be the unit
vector obtained by rotating $\nu$ on the other side of $\nu_{x}$ with the
angle $\frac{\pi}{2}-\frac{\theta}{2}$ and then
$\eta\cdot\nu=\cos(\frac{\pi}{2}-\frac{\theta}{2})>0$. The angle between
$\eta$ and $\nu_{x}$ is $\frac{\pi}{2}+\frac{\theta}{2}$ while
$\eta\cdot\nu_{x}=\cos(\frac{\pi}{2}+\frac{\theta}{2})<0$. This gives the
claim.

Via the claim above, we conclude that

$0<|Du(x)|(-\nu_{x})\cdot\eta=Du(x)\cdot\eta=\lim_{t\to
0^{+}}\frac{u(x+t\eta)-u(x)}{t},$

which implies that $u(x+t\eta)>u(x)$ for all small $t>0$. Therefore for such
$t>0$, one has $x+t\eta\in\Omega_{u(x)}$, which together with the definition
of $\nu$ yields that

$t\eta\cdot\nu=(x+t\eta-x)\cdot\nu\leq 0.$

This is a contradiction to our choice of $\eta$ which yields $\eta\cdot\nu>0$.
∎

Given any $x\in\Omega$, denote by $N^{p}_{x}$ the collection of all unit
vector $\nu$ so that

$\nu\cdot(y-x)\leq 0\quad\forall y\in\Omega^{p}_{u_{p}(x)}.$

Since $\Omega_{u_{p}(x)}^{p}$ is convex, then $N^{p}_{x}\neq\emptyset$. Given
any $\nu\in N^{p}_{x}$, recall $r_{x,\nu}$ as given in (4.1). Then
$x+r_{x,\nu}\nu\in\partial\Omega_{0}$, that is, $u_{p}(x+r_{x,\nu}\nu)=0$, and
moreover, $r_{x,\nu}$ also satisfies (4.2). As $u_{p}$ enjoys comparison with
cones from above (see Section 3), the following follows from an argument
similar to the one used to prove Lemma 4.1. We omit the details of the proof.

######  Lemma 4.2.

Let $p\in(2,\infty)$. For any $x\in\Omega$, it holds

$\displaystyle-
Du_{p}(x)\cdot\nu\geq\frac{u_{p}(x)-u_{p}(x+r\nu)}{r}>0\quad\forall\nu\in
N^{p}_{x}\quad r\in(0,r_{x,\nu}].$ (4.7)

Moreover,

$\displaystyle
N^{p}_{x}=\left\\{\nu^{p}_{x}:=-\frac{Du_{p}(x)}{|Du_{p}(x)|}\right\\}.$ (4.8)

Consequently,

$\displaystyle|Du_{p}(x)|\geq\frac{u_{p}(x)-u_{p}\left(x+r\nu\right)}{r}\quad\forall\nu\in
N_{x}^{p}\quad r\in(0,r_{x,\nu}].$ (4.9)

In particular,

$\displaystyle|Du_{p}(x)|\geq\frac{u_{p}(x)}{r_{x,\nu_{x}^{p}}}.$ (4.10)

######  Remark 4.3.

Thanks to (4.6) and (4.10), one concludes

$|Du(x)|\geq\frac{u(x)}{{\rm
diam}(\Omega_{0})}\quad\mbox{and}\quad|Du_{p}(x)|\geq\frac{u_{p}(x)}{{\rm
diam}(\Omega_{0})}\quad\forall x\in\Omega.$

This lower bound of $|Du|$ and $|Du_{p}|$ was given by Lindgren-Lindqvist
[44].

In this section, we finally give the following uniform lower bound and upper
bound of $|Du_{p}|$.

######  Lemma 4.4.

Given any ball $B(z,2r)\Subset\Omega$ we have

$\displaystyle\min_{\overline{B(z,r)}}|Du_{p}|\geq\frac{1}{{\rm
diam}(\Omega_{0})}\min\limits_{\overline{B(z,r)}}u_{p}\quad{\rm
and}\quad\max_{\overline{B(z,r)}}|Du_{p}|\leq\frac{1}{r}.$ (4.11)

Moreover, there exists a fixed constant $p_{z,r}>2$ such that

$\displaystyle\min_{\overline{B(z,r)}}|Du_{p}|\geq\frac{1}{2{\rm
diam}(\Omega_{0})}\min\limits_{\overline{B(z,r)}}u>0\quad\forall p>p_{z,r}.$
(4.12)

###### Proof.

Assume that $B(z,2r)\Subset\Omega$. Applying the inequality (4.10) in Lemma
4.2 we obtain

$\displaystyle\min_{\overline{B(z,r)}}|Du_{p}|\geq\frac{1}{{\rm
diam}(\Omega_{0})}\min\limits_{\overline{B(z,r)}}u_{p}\quad\forall p>2.$

Observe that $u_{p}\to u$ in $C^{0}(\overline{\Omega})$ and $u\neq 0$ on
$\overline{B(z,r)}$. Via $\min\limits_{\overline{B(z,r)}}u>0$ we can find a
fixed constant $p_{z,r}>2$ such that

$\min\limits_{\overline{B(z,r)}}u_{p}\geq\frac{1}{2}\min\limits_{\overline{B(z,r)}}u\quad\forall
p>p_{z,r}.$

This proves (4.12).

On the other hand, since $u_{p}$ enjoys the comparison with cones from above
(see Section 3), by Lemma 2.6 we have

$|Du_{p}(x)|={\mathop{\mathrm{missing}}{\,Lip}}\,u_{p}(x)\leq
S^{+}_{r}(u_{p},x)=\max_{|y-x|=r}\frac{u_{p}(y)-u_{p}(x)}{r}\quad\forall x\in
B(z,r).$

Thanks to $0\leq u_{p}(x)\leq 1$ and $B(z,2r)\Subset\Omega$, we obtain

$\max_{\overline{B(z,r)}}|Du_{p}|\leq\frac{1}{r}.$

Hence we complete this proof.

∎

##  5 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1, which is split into several
lemmas. The $C^{1}$ regularity of $u$ follows from Lemma 5.1. The locally
uniform convergence of $Du_{p}$ relies on Lemma 5.2.

Our first observation is continuity of $\nu_{x}=-\frac{Du(x)}{|Du(x)|}$ and
$|Du(x)|$ in $x\in\Omega$ coming from Lemma 4.1.

######  Lemma 5.1.

Suppose that $x_{k}\to x$ as $k\to\infty$. Then the following holds.

  * (i)

$\nu_{x_{k}}\to\nu_{x}$ as $k\to\infty$.

  * (ii)

$|Du(x_{k})|\to|Du(x)|$ as $k\to\infty$.

###### Proof.

Proof of (i). Let $\mu$ be one of limit points of $\\{\nu_{x_{k}}\\}$. Up to a
subsequence, we may assume that $\nu_{x_{k}}\to\mu$. Below we show that
$\mu=\nu_{x}$. Given any $y\in\Omega_{u(x)}$, since $u(x_{k})\to u(x)$, for
all sufficiently large $k$ we have $y\in\Omega_{u(x_{k})}$, and hence

$\nu_{x_{k}}\cdot(y-x_{k})\leq 0.$

Letting $k\to\infty$ one has

$\mu\cdot(y-x)\leq 0.$

Then by (4.4) in Lemma 4.1 we have $\mu=\nu_{x}$. This proves (i).

Proof of (ii). We already know that
$|Du(x)|\geq\limsup_{k\to\infty}|Du(x_{k})|$ by the upper semicontinuous of
$|Du|$; see [17]. Thus it suffices to prove

$\displaystyle\liminf_{k\to\infty}|Du(x_{k})|\geq|Du(x)|.$ (5.1)

Recall that (4.5) in Lemma 4.1 give us

$|Du(x_{k})|\geq\frac{u(x_{k})-u\left(x_{k}+t\nu_{x_{k}}\right)}{t}\quad\forall
k\geq 1,\quad\forall 0<t<\frac{1}{4}{\rm dist}(x_{k},\partial\Omega_{0}).$

Sending $k\to\infty$, by $\nu_{x_{k}}\to\nu_{x}$ in the proof of (i) and $u\in
C^{0}(\Omega_{0})$ one gets

$u(x_{k})-u\left(x_{k}+t\nu_{x_{k}}\right)\to
u(x)-u(x+t\nu)\quad\mbox{and}\quad{\rm dist}(x_{k},\partial\Omega_{0})\to{\rm
dist}(x,\partial\Omega_{0}),$

and hence

$\displaystyle\liminf_{k\to\infty}|Du(x_{k})|\geq\frac{u(x)-u\left(x+t\nu_{x}\right)}{t}\quad\forall
0<t<\frac{1}{4}{\rm dist}(x,\partial\Omega_{0}).$

Observe that

$\displaystyle u\left(x+t\nu_{x}\right)-u(x)=-t|Du(x)|+o(t)$

and hence

$\frac{u(x)-u\left(x+t\nu_{x}\right)}{t}=|Du(x)|+o(1).$

As $t\to 0$, it follows that (5.1). Hence we complete this proof. ∎

To obtain the locally uniform convergence of $Du_{p}$, we also need the
following lemma, which is exactly the same as Lemma 5.1.

######  Lemma 5.2.

Suppose that $x_{j}\to x\in\Omega$ and $p_{j}\to\infty$ as $j\to\infty$. Then
the following holds.

  * (i)

$\nu^{p_{j}}_{x_{j}}\to\nu_{x}$ as $j\to\infty$.

  * (ii)

$|Du_{p_{j}}(x_{j})|\to|Du(x)|$ as $j\to\infty$.

###### Proof.

Proof of (i). Thanks to Lemma 4.4, up to a subsequence, we may assume that
$\nu^{p_{j}}_{x_{j}}$ converges to some unit vector $\mu$. It is enough to
show that $\mu=\nu_{x}$. Due to Lemma 4.1, we only need to show that $\mu\in
N_{x}$, that is,

$\mu\cdot(y-x)\leq 0\quad\forall y\in\Omega_{u(x)}.$

Given any $y\in\Omega_{u(x)}$, one has $u(y)>u(x)$. Recall that $u_{p_{j}}\to
u$ uniformly in $\overline{\Omega}$ and $x_{j}\to x\in\Omega$ as $j\to\infty$,
we know that $u_{p_{j}}(x_{j})\to u(x)$, $u_{p_{j}}(y)\to u(y)$ and
$u_{p_{j}}(x_{j})\to u(x)$ as $j\to\infty$. For sufficiently large $j$, it
then follows $u_{p_{j}}(y)>u_{p_{j}}(x_{j})$, that is,
$y\in\Omega^{p_{j}}_{u_{j}(x)}.$ By Lemma 4.2, one gets

$\displaystyle\nu^{p_{j}}_{x_{j}}\cdot(y-x_{j})\leq 0.$

Letting $j\to\infty$, we conclude that $\mu\cdot(y-x)\leq 0$ as desired.

Proof of (ii). Since $x_{j}\to x\in\Omega$, via Lemma 4.4 we know that
$\\{|Du_{p_{j}}(x_{j})|\\}_{j}$ is bounded uniformly $p_{j}>2$.

We first show $\limsup_{j\to\infty}|Du_{p_{j}}(x_{j})|\leq|Du(x)|$. Indeed,
since $x_{j}\to x$ as $j\to\infty$, for all sufficiently large $j$ so that
$\frac{1}{2}{\rm dist}(x,\partial\Omega)\leq{\rm dist}(x_{j},\partial\Omega)$.
For such $j$, given any $0<r<\frac{1}{2}{\rm dist}(x,\partial\Omega)$, by
Lemma 2.6 and Lemma 3.2, we have $|Du_{p_{j}}(x_{j})|\leq
S^{+}_{r}(u_{p_{j}},x_{j})$. Choose $y_{j,r}\in\partial B(x_{j},r)$ so that

$S^{+}_{r}(u_{p_{j}},x_{j})=\max_{|y-x_{j}|=r}\frac{u_{p_{j}}(y)-u_{p_{j}}(x_{j})}{r}=\frac{u_{p_{j}}(y_{j,r})-u_{p_{j}}(x_{j})}{r}.$

We then have

$|Du_{p_{j}}(x_{j})|\leq\frac{u_{p_{j}}(y_{j,r})-u_{p_{j}}(x_{j})}{r}.$

Noting that

$r-|x_{j}-x|=|y_{j,r}-x_{j}|-|x_{j}-x|\leq|y_{j,r}-x|\leq|y_{j,r}-x_{j}|+|x_{j}-x|=r+|x_{j}-x|,$

one has $\lim_{j\to\infty}|y_{j,r}-x|=r$, and hence, all limits of $y_{j,r}$
belong to $\partial B(x,r)$. Since $u_{p_{j}}\to u$ uniformly in
$\overline{\Omega}$, we get

$\displaystyle\limsup_{j\to\infty}|Du_{p_{j}}(x_{j})|$
$\displaystyle=\limsup_{j\to\infty}\frac{u_{p_{j}}(y_{j,r})-u_{p_{j}}(x_{j})}{r}\leq\sup_{y\in\partial
B(x,r)}\frac{u(y)-u(x)}{r}=S^{+}_{r}(u,x).$

Therefore, by Lemma 2.6 and Theorem 2.8, one has

$\limsup_{j\to\infty}|Du_{p_{j}}(x_{j})|\leq\lim_{r\to
0}S^{+}_{r}(u,x)=|Du(x)|.$

On the other hand, by (4.9) in Lemma 4.2 we have

$|Du_{p_{j}}(x_{j})|\geq\frac{u_{p_{j}}(x_{j})-u_{p_{j}}\left(x_{j}+t\nu_{x_{j}}^{p_{j}}\right)}{t}\quad\forall
j\geq 1,\quad\forall t<\frac{1}{4}{\rm dist}(x_{j},\partial\Omega_{0}).$

Since $u_{p_{j}}(x_{j})\to u(x)$ and $\nu_{x_{j}}^{p_{j}}\to\nu_{x}$ due to
(i) in this lemma, letting $j\to\infty$ we get

$\liminf_{j\to\infty}|Du_{p_{j}}(x_{j})|\geq\frac{u(x)-u\left(x+t\nu_{x}\right)}{t},\quad\forall
0<t<\frac{1}{4}{\rm dist}(x,\partial\Omega_{0}).$

Using the differentiability of $u$ in $\Omega$, by sending $t\to 0$ we
conclude

$\liminf_{j\to\infty}|Du_{p_{j}}(x_{j})|\geq-Du(x)\cdot\nu_{x}=|Du(x)|.$

Hence we finish this proof. ∎

Let us now proof Theorem 1.1.

###### Proof of Theorem 1.1.

The $C^{1}$ regularity of $u$ follows from Lemma 5.1. Also, the locally
uniform convergence of $Du_{p}\to Du$ is a direct consequence of Lemma 5.2. ∎

##  6 Two fundamental inequalities under quasi-concavity

We establish a fundamental algebraic and geometric structural inequalities
involving $\Delta_{\infty}v$ for smooth quasi-concave functions $v$ via the
geometry of the contour surfaces.

######  Lemma 6.1.

Let $n\geq 2$ and let $v:{{{\mathbb{R}}}^{n}}\to{\mathbb{R}}$ be a quasi-
concave function. If $v\in C^{\infty}(U)$ for some domain
$U\subset{{{\mathbb{R}}}^{n}}$, then

$\displaystyle 2[|D^{2}vDv|^{2}-\Delta
v\Delta_{\infty}v]\geq|Dv|^{2}[|D^{2}v|^{2}-(\Delta v)^{2}]\quad{\rm in}\quad
U.$ (6.1)

This further allows us to get the following fundamental divergence inequality
for quasi-concave functions.

######  Lemma 6.2.

Let $n\geq 2$ and let $v:{{{\mathbb{R}}}^{n}}\to{\mathbb{R}}$ be a quasi-
concave function. If $v\in C^{\infty}(U)$ for some domain
$U\subset{{{\mathbb{R}}}^{n}}$, then

$\displaystyle{\rm div}\left(|Dv|^{-2}(\Delta vDv-D^{2}vDv)\right)\geq
0\quad{\rm in}\quad U\backslash\\{Dv\neq 0\\}.$

######  Remark 6.3.

In dimension $n=2$, for any function $v\in C^{\infty}(U)$ it was shown in [39]
that

$2[|D^{2}vDv|^{2}-\Delta v\Delta_{\infty}v]=|Dv|^{2}[|D^{2}v|^{2}-(\Delta
v)^{2}]\quad{\rm in}\quad U$

and

${\rm div}\left(|Dv|^{-2}(\Delta vDv-D^{2}vDv)\right)=0\quad{\rm in}\quad
U\backslash\\{Dv\neq 0\\}.$

So the main ingredient for Lemma 6.1 and Lemma 6.2 is in dimension $n\geq 3$.

We need the following lemma to prove Lemma 6.1. We denote $e_{n}=(0,...,0,1)$.

######  Lemma 6.4.

Let $n\geq 3$ and let $v:{{{\mathbb{R}}}^{n}}\to{\mathbb{R}}$ be a quasi-
concave function. Assume that $v\in C^{2}(B(0,r))$, $v(0)=0$ and
$Dv(0)=e_{n}$. Then

$\displaystyle(w_{x_{i}x_{j}})^{2}-w_{x_{i}x_{i}}w_{x_{j}x_{j}}\leq
0\quad\forall 1\leq i<j\leq n-1.$ (6.2)

The inequality (6.2) essentially says that the sectional curvature is non-
positive at $0$. This is well-known in differential geometry. For reader’s
convenience, we give the details via an analysis argument.

###### Proof.

First, note that the quasi-concavity of $v$ guarantees

$v(y)\leq 0$ for all $y\in B(0,s)$ with $y_{n}=0$ (6.3)

for some $0<s<r$. Indeed, since $v\in C^{2}(B(0,r))$ and $Dv(0)=e_{n}$, by the
implicit theorem we know that the equation $v(y)=0$ determines a
$(n-1)$-dimension surface $S$ in a ball $B(0,s)$ for some $0<s<r$. The
boundary of $\\{y\in B(0,s)|v(y)>0\\}$ is $(S\cap B(0,s))\cup(\partial
B(0,s)\cap\\{v>0\\})$. Since $Dv(0)=e_{n}$, the tangential plane of $S$ at $0$
is given by the hyperplane $P=\\{y\in{{{\mathbb{R}}}^{n}}|y_{n}=0\\}$. The
convexity of $\\{y\in{{{\mathbb{R}}}^{n}}|v(y)>0\\}$ implies that the
convexity of $\\{y\in B(0,s)|v(y)>0\\}$. Thus $\\{y\in B(x,s)|v(y)>0\\}$ lies
in above $P$, that is, if $v(y)>0$, then $y_{n}>0$. Therefore, (6.3) holds as
desired.

Applying (6.3), we claim that

$\displaystyle D^{2}v(0)\xi\cdot\xi\leq
0\quad\forall\xi\in{{{\mathbb{R}}}^{n}}\quad{\rm with}\quad\xi_{n}=0,$ (6.4)

that is, the second fundamental form of $S$ at point $0$ is non-positive
definite. Indeed, given any $\xi\in\partial B(0,1)$, by Taylor’s expansion and
$Dv(0)=e_{n}$ one has

$v(t\xi)=v(0)+t\xi_{n}+\frac{t^{2}}{2}D^{2}v(0)\xi\cdot\xi+o(t^{2}).$

If $\xi_{n}=0$, by (6.3) one then has $v(t\xi)\leq v(0)=0$ and hence

$D^{2}v(0)\xi\cdot\xi=\lim_{t\to 0}\frac{2}{t^{2}}[v(t\xi)-v(0)]\leq 0.$

This gives the claim (6.4).

Next, at point $0$ we apply (6.4) to get

$\displaystyle(w_{x_{i}x_{j}})^{2}-w_{x_{i}x_{i}}w_{x_{j}x_{j}}\leq
0\quad\forall 1\leq i<j\leq n-1.$

Indeed, given any pair $(i,j)$ with $1\leq i<j\leq n-1$, write

$Q:=\begin{bmatrix}w_{x_{i}x_{i}}&\ w_{x_{i}x_{j}}\\\ w_{x_{j}x_{i}}&\
w_{x_{j}x_{j}}\end{bmatrix}.$

Applying (6.4) with $\xi=ae_{i}+be_{j}$ for all $a,b\in{\mathbb{R}}$ we obtain

$(a,b)Q(a,b)^{T}=a^{2}w_{x_{i}x_{i}}+b^{2}w_{x_{j}x_{j}}+2abw_{x_{i}x_{j}}=\xi^{T}D^{2}w(0)\xi\leq
0.$

Clearly, this quadratic form is negative semi-definite, and hence its
eigenvalues of $Q$ are non-positive. Thus

$\det Q=w_{x_{i}x_{i}}w_{x_{j}x_{j}}-(w_{x_{i}x_{j}})^{2}\geq 0,$

which gives (6.2). ∎

We are ready to prove Lemma 6.1.

###### Proof of Lemma 6.1.

By Remark 6.3, we only need to consider the dimension $n\geq 3$. Assume that
$v:{{{\mathbb{R}}}^{n}}\to{\mathbb{R}}$ is a quasi-concave function and $v\in
C^{\infty}(U)$. Fix any $x\in U$. Up to considering $v(z+x)-v(x)$ with $z+x\in
U$ we may assume that $x=0\in U$ and $v(0)=0$. If $Dv(0)=0$, then (6.1) holds
trivially. So we assume $Dv(0)\neq 0$ below. Up to considering $v/|Dv(0)|$, we
may assume that $|Dv(0)|=1$. Below we consider the following two cases:
$Dv(0)=e_{n}$, and $Dv(0)\neq e_{n}$.

Case $Dv(0)=e_{n}$. At the point $0$, by $Dv=e_{n}$, a direct calculation
gives

$\Delta_{\infty}v=v_{x_{n}x_{n}},\quad|D^{2}vDv|^{2}=\sum_{1\leq i\leq
n}(v_{x_{i}x_{n}})^{2}.$

We then write (6.1) as

$2\sum_{1\leq i\leq n}(v_{x_{i}x_{n}})^{2}-2\Delta
vv_{x_{n}x_{n}}\geq|D^{2}v|^{2}-(\Delta v)^{2},$

that is,

$\displaystyle|D^{2}v|^{2}\leq(\Delta v)^{2}-2\Delta
vv_{x_{n}x_{n}}+2\sum_{1\leq i\leq n}(v_{x_{i}x_{n}})^{2}.$ (6.5)

To prove (6.5), we write

$\displaystyle|D^{2}v|^{2}$
$\displaystyle=\sum_{i=1}^{n}\left(v_{x_{i}x_{i}}\right)^{2}+2\sum_{1\leq
i<j\leq n}\left(v_{x_{i}x_{j}}\right)^{2}$
$\displaystyle=\sum_{i=1}^{n-1}\left(v_{x_{i}x_{i}}\right)^{2}+(v_{x_{n}x_{n}})^{2}+2\sum_{1\leq
i<j\leq n-1}\left(v_{x_{i}x_{j}}\right)^{2}+2\sum_{1\leq i\leq
n-1}(v_{x_{i}x_{n}})^{2}.$

Note that

$\displaystyle\sum_{i=1}^{n-1}\left(v_{x_{i}x_{i}}\right)^{2}$
$\displaystyle=\left(\sum_{i=1}^{n-1}v_{x_{i}x_{i}}\right)^{2}-2\sum_{1\leq
i<j\leq n-1}v_{x_{i}x_{i}}v_{x_{j}x_{j}}$ $\displaystyle=(\Delta
v-v_{x_{n}x_{n}})^{2}-2\sum_{1\leq i<j\leq n-1}v_{x_{i}x_{i}}v_{x_{j}x_{j}}$
$\displaystyle=(\Delta v)^{2}-2\Delta
vv_{x_{n}x_{n}}+(v_{x_{n}x_{n}})^{2}-2\sum_{1\leq i<j\leq
n-1}v_{x_{i}x_{i}}v_{x_{j}x_{j}}$

and

$2\sum_{1\leq i\leq n-1}(v_{x_{i}x_{n}})^{2}=2\sum_{1\leq i\leq
n}(v_{x_{i}x_{n}})^{2}-2(v_{x_{n}x_{n}})^{2}.$

We therefore obtain

$\displaystyle|D^{2}v|^{2}$ $\displaystyle=(\Delta v)^{2}-2\Delta
vv_{x_{n}x_{n}}+2(v_{x_{n}x_{n}})^{2}$ $\displaystyle\quad\quad+2\sum_{1\leq
i<j\leq n-1}[(v_{x_{i}x_{j}})^{2}-v_{x_{i}x_{i}}v_{x_{j}x_{j}}]+2\sum_{1\leq
i\leq n}(v_{x_{i}x_{n}})^{2}-2(v_{x_{n}x_{n}})^{2}$ $\displaystyle=(\Delta
v)^{2}-2\Delta vv_{x_{n}x_{n}}+2\sum_{1\leq i\leq
n}(v_{x_{i}x_{n}})^{2}+2\sum_{1\leq i<j\leq
n-1}[(v_{x_{i}x_{j}})^{2}-v_{x_{i}x_{i}}v_{x_{j}x_{j}}].$

Since Lemma 6.4 tells us that

$\sum_{1\leq i<j\leq
n-1}[(v_{x_{i}x_{j}})^{2}-v_{x_{i}x_{i}}v_{x_{j}x_{j}}]\leq 0,$

we get (6.5) as desired.

Case $Dv(0)\neq e_{n}$. Denote by $O$ the orthogonal matrix so that
$Dv(0)=Oe_{n}$. Define

$w(y):=v(Oy)\quad\forall y\in U.$

At point $0$, we have $Dw=O^{T}Dv=e_{n}$. Applying (6.5) to $w$, we get

$2[|D^{2}wDw|^{2}-\Delta w\Delta_{\infty}w]\geq[|D^{2}w|^{2}-(\Delta w)^{2}].$

This would imply (6.1) for $v$ at $0$ once we have

$|D^{2}wDw|^{2}=|D^{2}vDv|^{2}$, $|D^{2}w|^{2}=|D^{2}v|^{2}$, $\Delta w=\Delta
v$ and $\Delta_{\infty}w=\Delta v_{\infty}$ at 0.

We check this as below. At point $0$, we have

$D^{2}w=O^{T}D^{2}vO$ and $D^{2}wDw=O^{T}D^{2}v(OO^{T})Dv=O^{T}D^{2}vDv.$

Thus $|D^{2}w|^{2}=|D^{2}v|^{2}$ and $|D^{2}wDw|^{2}=|D^{2}vDv|^{2}.$ By
$OO^{T}=I_{n}$, we also have

$\displaystyle\Delta_{\infty}w$
$\displaystyle=(Dw)^{T}D^{2}wDw=(Dv)^{T}(OO^{T})D^{2}vDv=(Dv)^{T}D^{2}vDv=\Delta_{\infty}v.$

Moreover, the cyclic property of trace implies that

$\displaystyle\Delta w$ $\displaystyle={\rm tr}(D^{2}w)={\rm
tr}(O^{T}D^{2}vO)={\rm tr}(OO^{T}D^{2}v)={\rm tr}(D^{2}v)=\Delta v.$

The proof is complete. ∎

Finally we apply Lemma 6.1 to prove Lemma 6.2.

###### Proof of Lemma 6.2.

At any point where $Dv\neq 0$, a direct calculation gives

$\displaystyle{\rm div}(|Dv|^{-2}\Delta vDv)=-2|Dv|^{-4}\Delta_{\infty}v\Delta
v+|Dv|^{-2}(\Delta v)^{2}+|Dv|^{-2}D\Delta v\cdot Dv$

and

$\displaystyle{\rm
div}(|Dv|^{-2}D^{2}vDv)=-2|Dv|^{-4}|D^{2}vDv|^{2}+|Dv|^{-2}D\Delta v\cdot
Dv+|Dv|^{-2}|D^{2}v|^{2}.$

Combining them and using Lemma 6.1 one then obtains

$\displaystyle{\rm div}\left(|Dv|^{-2}\Delta vDv-|Dv|^{-2}D^{2}vDv\right)$
$\displaystyle=2|Dv|^{-4}[|D^{2}vDv|^{2}-\Delta
v\Delta_{\infty}v]+|Dv|^{-2}[(\Delta v)^{2}-|D^{2}v|^{2}]\geq 0$

as desired. The proof is complete. ∎

##  7 Proof of Theorem 1.3

In this section we prove Theorem 1.3. Firstly, via Lemma 6.2 we established
the following upper bound for $D|Du_{p}|^{\alpha}$ with $\alpha<0$, which is
uniform in all $p\in[2,\infty)$. Recall that $u_{p}$ is the $p$-harmonic
potential and $u$ is the $\infty$-harmonic potential in the convex ring
$\Omega$.

######  Lemma 7.1.

Given any $\alpha<0$, for any $p\in(2,\infty)$ we have

$\displaystyle\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx\leq
C(1+|\alpha|^{2})r^{-2}\int_{B(z,r)}|Du_{p}|^{2\alpha}\,dx\quad\mbox{whenever}\quad
B(z,r)\Subset\Omega,$ (7.1)

where $C$ is a universal constant.

###### Proof.

Since $Du_{p}\neq 0$ on $\Omega$, applying Lemma 6.2 to $u_{p}$ one has

$\displaystyle{\rm div}\left(|Du_{p}|^{-2}[\Delta
u_{p}Du_{p}-D^{2}u_{p}Du_{p}]\right)\geq 0\quad{\rm in}\quad
B(z,r)\Subset\Omega.$

Multiplying both sides by a test function $\xi^{2}|Du_{p}|^{2\alpha}$ with
$\alpha<0$ and $\xi\in C^{\infty}_{c}(B(z,r))$, one gets

$\displaystyle\int_{B(z,r)}{\rm div}\left(|Du_{p}|^{-2}[\Delta
u_{p}Du_{p}-D^{2}u_{p}Du_{p}]\right)\xi^{2}|Du_{p}|^{2\alpha}\,dx\geq 0.$

Via integration by parts, it becomes

$\displaystyle-2\alpha\int_{B(z,r)}|Du_{p}|^{2\alpha-4}|D^{2}u_{p}Du_{p}|^{2}\xi^{2}\,dx+2\alpha\int_{B(z,r)}|Du_{p}|^{2\alpha-4}\Delta
u_{p}\Delta_{\infty}u_{p}\xi^{2}\,dx$
$\displaystyle\leq-2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}\Delta u_{p}Du_{p}\cdot
D\xi\xi\,dx+2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}D^{2}u_{p}Du_{p}\cdot
D\xi\xi\,dx.$ (7.2)

For the second term in the left-hand side of (7), using
$-\Delta^{N}_{\infty}u_{p}=\frac{1}{p-2}\Delta u_{p}$ in $\Omega$, by
$\alpha<0$ and $p>2$ we have

$2\alpha\int_{B(z,r)}|Du_{p}|^{2\alpha-4}\Delta
u_{p}\Delta_{\infty}u_{p}\xi^{2}\,dx=-\frac{2\alpha}{p-2}\int_{B(z,r)}|Du_{p}|^{2\alpha-2}(\Delta
u_{p})^{2}\xi^{2}\,dx\geq 0.$

For the first term of right-hand side of (7), by integration by parts again
and using $\alpha<0$, we obtain

$\displaystyle-2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}\Delta u_{p}Du_{p}\cdot
D\xi\xi\,dx$
$\displaystyle=2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}D^{2}u_{p}Du_{p}\cdot
D\xi\xi\,dx+4(\alpha-1)\int_{B(z,r)}|Du_{p}|^{2\alpha-4}\Delta_{\infty}u_{p}Du_{p}\cdot
D\xi\xi\,dx$ $\displaystyle\quad+2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}D^{2}\xi
Du_{p}\cdot Du_{p}\xi\,dx+2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}(Du_{p}\cdot
D\xi)^{2}\,dx$
$\displaystyle\leq-\frac{\alpha}{4}\int_{B(z,r)}|Du_{p}|^{2\alpha-4}|D^{2}u_{p}Du_{p}|^{2}\xi^{2}\,dx+C[1+|\alpha|+\frac{1}{|\alpha|}]\int_{B(z,r)}|Du_{p}|^{2\alpha}(|D\xi|^{2}+|D^{2}\xi||\xi|)\,dx,$

where we also used the Young’s inequality and Cauchy-Schwartz’ inequality in
the last inequality.

For the second term of right-hand side of (7), it follows by Young’s
inequality that

$\displaystyle 2\int_{B(z,r)}|Du_{p}|^{2\alpha-2}D^{2}u_{p}Du_{p}\cdot
D\xi\xi\,dx$
$\displaystyle\leq-\frac{\alpha}{4}\int_{B(z,r)}|Du_{p}|^{2\alpha-4}|D^{2}u_{p}Du_{p}|^{2}\xi^{2}\,dx+C\frac{1}{|\alpha|}\int_{B(z,r)}|Du_{p}|^{2\alpha}|D\xi|\,dx.$

Combining above we have

$-\frac{\alpha}{2}\int_{B(z,r)}|Du_{p}|^{2\alpha-4}|D^{2}u_{p}Du_{p}|^{2}\xi^{2}\,dx\leq
C[1+|\alpha|+\frac{1}{|\alpha|}]\int_{B(z,r)}|Du_{p}|^{2\alpha}(|D\xi|^{2}+|D^{2}\xi||\xi|)\,dx.$

Since $Du_{p}\neq 0$ on $B(z,r)$, rewriting

$|Du_{p}|^{2\alpha-4}|D^{2}u_{p}Du_{p}|^{2}=\frac{1}{\alpha^{2}}|D|Du_{p}|^{\alpha}|^{2},$

we deduce

$\displaystyle\int_{B(z,r)}|D|Du_{p}|^{\alpha}|^{2}\xi^{2}\,dx\leq
C(1+|\alpha|^{2})\int_{B(z,r)}|Du_{p}|^{2\alpha}(|D\xi|^{2}+|D^{2}\xi||\xi|)\,dx.$

Choosing a suitable cut-off function $\xi\in C^{\infty}_{c}(B(z,r))$ we obtain

$\displaystyle\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx\leq
C(1+|\alpha|^{2})r^{-2}\int_{B(z,r)}|Du_{p}|^{2\alpha}\,dx$

as desired. ∎

Consequently, we have the following upper bound for $D|Du_{p}|^{\alpha}$ with
$\alpha\geq 0$, which is uniform in $p>2$. Recall that when $\alpha=0$,
$f^{\alpha}$ is understood as $\ln f$.

######  Lemma 7.2.

Given any $\alpha\geq 0$, for any $p\in(2,\infty)$ we have

$\displaystyle\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx\leq
C\alpha^{2}r^{-2}\max_{B(z,r/2)}|Du_{p}|^{2\alpha+2}\int_{B(z,r)}|Du_{p}|^{-2}\,dx\quad\mbox{whenever
$B(z,r)\Subset\Omega$,}$

where $C$ is a universal constant.

###### Proof.

Let $B(z,r)\Subset\Omega$ and $\alpha\geq 0$. Since $Du_{p}\neq 0$ in
$\Omega$, one has

$|D|Du_{p}|^{\alpha}|=|D(|Du_{p}|^{-1})^{-\alpha}|=\alpha(|Du_{p}|^{-1})^{-\alpha-1}|D|Du_{p}|^{-1}|=\alpha|Du_{p}|^{1+\alpha}|D|Du_{p}|^{-1}|.$

Then we conclude from (7.1) in Lemma 7.1 that

$\displaystyle\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx$
$\displaystyle\leq\alpha^{2}\max_{B(z,r/2)}|Du_{p}|^{2\alpha+2}\int_{B(z,r/2)}|D|Du_{p}|^{-1}|^{2}\,dx$
$\displaystyle\leq
C\alpha^{2}r^{-2}\max_{B(z,r/2)}|Du_{p}|^{2\alpha+2}\int_{B(z,r)}|Du_{p}|^{-2}\,dx.$

Since

$|D\ln|Du_{p}||=|D\ln|Du_{p}|^{-1}|=|Du||D|Du_{p}|^{-1}|,$

it follows from (7.1) in Lemma 7.1 again that

$\displaystyle\int_{B(z,r/2)}|D\ln|Du_{p}||^{2}\,dx$
$\displaystyle\leq\max_{B(z,r)}|Du_{p}|^{2}\int_{B(z,r/2)}|D|Du_{p}|^{-1}|^{2}\,dx$
$\displaystyle\leq
Cr^{-2}\max_{B(z,r/2)}|Du_{p}|^{2}\int_{B(z,r)}|Du_{p}|^{-2}\,dx.$

∎

Thanks to the local uniform bound of $|Du_{p}|$ in Lemma 4.4, sending
$p\to\infty$, we get the following.

######  Lemma 7.3.

It holds that $|Du|^{\alpha}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ for any
$\alpha\in{\mathbb{R}}$ and $|Du_{p}|^{\alpha}\to|Du|^{\alpha}$ weakly in
$W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ for any
$\alpha\in{\mathbb{R}}$ as $p\to\infty$. Moreover, we have the following
quantitative upper bound. If $\alpha<0$, then

$\displaystyle\int_{B(z,r/2)}|D|Du|^{\alpha}|^{2}\,dx\leq
C(1+|\alpha|^{2})r^{-2}\int_{B(z,r)}|Du|^{2\alpha}\,dx\quad\mbox{whenever}\quad
B(z,r)\Subset\Omega.$

If $\alpha\geq 0$, then

$\displaystyle\int_{B(z,r/2)}|D|Du|^{\alpha}|^{2}\,dx\leq
C\alpha^{2}r^{-2}\max_{B(z,r/2)}|Du|^{2\alpha+2}\int_{B(z,r)}|Du|^{-2}\,dx\quad\mbox{whenever}\quad
B(z,r)\Subset\Omega.$

Here $C$ are universal constants.

###### Proof.

We only consider the case $\alpha<0$. The case $\alpha\geq 0$ is similar; we
omit the details.

For any $B(z,r)\Subset\Omega$ we recall from Lemma 7.1 that

$\displaystyle\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx$ $\displaystyle\leq
C(1+\alpha^{2})r^{-2}\int_{B(z,r)}|Du_{p}|^{2\alpha},$

where $C$ is a universal constant. By (4.11) and (4.12) in Lemma 4.4, we have
$|Du_{p}|^{\alpha}\in W^{1,2}(B(z,r/2))$ uniformly $p>2$. Using the weak
compactness of Sobolev space $W^{1,2}(B(z,r/2))$(see [28, Section 5.7]), there
exists a function $g_{\alpha}\in W^{1,2}(B(z,r/2))$ such that

$\displaystyle D|Du_{p}|^{\alpha}\to Dg_{\alpha}\ {\rm weakly\ in}\
L^{2}(B(z,r))\ {\rm and}\ |Du_{p}|^{\alpha}\to g_{\alpha}\ {\rm in}\
L^{2}(B(z,r))\mbox{ as $p\to\infty$.}$

Since Theorem 1.1 gives $Du_{p}\to Du$ in $C^{1}(\Omega)$, via Lemma 4.4 we
know that

$|Du_{p}|^{\alpha}\to|Du|^{\alpha}$ in $C^{0}(\Omega)$ for any
$\alpha\in{\mathbb{R}}\setminus\\{0\\}$.

Therefore $g_{\alpha}=|Du|^{\alpha}$ and then

$\displaystyle\int_{B(z,r/2)}|D|Du|^{\alpha}|^{2}\,dx$
$\displaystyle\leq\liminf_{p\to\infty}\int_{B(z,r/2)}|D|Du_{p}|^{\alpha}|^{2}\,dx$
$\displaystyle\leq
C(1+\alpha^{2})r^{-2}\liminf_{p\to\infty}\int_{B(z,r)}|Du_{p}|^{2\alpha}\,dx$
$\displaystyle=C(1+\alpha^{2})r^{-2}\int_{B(z,r)}|Du|^{2\alpha}\,dx$

as desired. ∎

As a direct consequence of Lemma 7.3, we show that the partial derivative of
$|Du|^{\alpha}$ along $Du$ is zero.

######  Lemma 7.4.

For any $\alpha\in{\mathbb{R}}$, we have $D|Du|^{\alpha}\cdot Du=0$ almost
everywhere in $\Omega$.

###### Proof.

By $Du_{p}\neq 0$ in $\Omega$, we note that

$D\ln|Du_{p}|\cdot
Du_{p}=D|Du_{p}|\cdot\frac{Du_{p}}{|Du_{p}|}=\Delta_{\infty}^{N}u_{p}=-\frac{1}{p-2}\Delta
u_{p}\quad{\rm in}\quad\Omega.$

For any $\phi\in C^{\infty}_{c}(\Omega)$, it then follows that

$\int_{\Omega}D\ln|Du_{p}|\cdot
Du_{p}\phi\,dx=-\frac{1}{p-2}\int_{\Omega}\Delta
u_{p}\phi\,dx=\frac{1}{p-2}\int_{\Omega}Du_{p}\cdot D\phi\,dx.$

Since $Du_{p}\to Du$ in $C^{0}(\Omega)$ by Theorem 1.1 and
$\ln|Du_{p}|\to\ln|Du|$ weakly in
$W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ by Lemma 7.3, we have

$\int_{\Omega}D\ln|Du|\cdot Du\phi\,dx=0.$

Thus $D\ln|Du|\cdot Du$=0 almost everywhere in $\Omega$.

Recall that $u\in C^{1}(\Omega)$ and $Du\neq 0$ in $\Omega$ according to
Theorem 1.1. For any $\alpha\in{\mathbb{R}}\setminus\\{0\\}$, it follows from
$|Du|^{\alpha}\in W^{1,2}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ that

$D|Du|^{\alpha}=|Du|^{\alpha}D\ln|Du|$ almost everyhwere in $\Omega$.

Therefore $D|Du|^{\alpha}\cdot Du$=0 almost everywhere in $\Omega$, which is
as desired. ∎

Now we can finish the proof of Theorem 1.3.

###### Proof of Theorem 1.3.

It follows from Lemma 7.3 and Lemma 7.4. ∎

##  8 Proof of Theorem 1.4

In this section, we assume that $u$ is the $\infty$-harmonic potential in
$\Omega=\Omega_{0}\setminus\\{x_{0}\\}$, where $\Omega_{0}$ is a convex domain
and $\overline{\Omega}_{1}=\\{x_{0}\\}$ with $x_{0}\in\Omega_{0}$.

To prove Theorem 1.4, we need the following key lemma, which when $n=2$ was
already proved by Lindgren-Lindqvist [43, Corollary 10].

######  Lemma 8.1.

For any $x\in\Omega$,

$|Du(x)|\leq\frac{1}{{\mathop{\mathrm{missing}}{\,dist\,}}(x,\partial\Omega_{0})}.$

Moreover, we have

$\lim_{x\to
x_{0}}|Du(x)|=\|Du\|_{L^{\infty}(\Omega_{0})}=\frac{1}{{\rm{\mathop{\mathrm{missing}}{\,dist\,}}}(x_{0},\partial\Omega_{0})}.$

###### Proof.

Thanks to the convexity of $\Omega_{0}$, we deduce

$\|Du\|_{L^{\infty}(\Omega_{0})}=\sup_{x\in\Omega_{0}}{\mathop{\mathrm{missing}}{\,Lip}}\,u(x)={\mathop{\mathrm{missing}}{\,Lip}}(u,\Omega_{0}):=\sup_{x\neq
y,x,y\in\Omega_{0}}\frac{u(x)-u(y)}{|x-y|}.$

From $\Omega=\Omega_{0}\setminus\\{x_{0}\\}$ and $u\in C^{0}(\Omega_{0})$, it
follows that
${\mathop{\mathrm{missing}}{\,Lip}}(u,\Omega_{0})={\mathop{\mathrm{missing}}{\,Lip}}(u,\Omega)$.
Since $u$ is an absolutely minimizing Lipschitz extension, we know

${\mathop{\mathrm{missing}}{\,Lip}}(u,\Omega)={\mathop{\mathrm{missing}}{\,Lip}}(u,\partial\Omega)=\sup_{y\in\partial\Omega_{0}}\frac{1}{|x_{0}-y|}=\frac{1}{{\mathop{\mathrm{missing}}{\,dist\,}}(x_{0},\partial\Omega_{0})}.$

Therefore by Theorem 2.8,

$|Du(x)|={\mathop{\mathrm{missing}}{\,Lip}}\,u(x)\leq\|Du\|_{L^{\infty}(\Omega_{0})}=\frac{1}{{\mathop{\mathrm{missing}}{\,dist\,}}(x_{0},\partial\Omega_{0})}\quad\forall
x\in\Omega.$

Below we show that $\liminf_{x\to
x_{0}}|Du(x)|\geq\|Du\|_{L^{\infty}(\Omega_{0})}$. To this end, using Theorem
1.1 in [58] we have

$\lim_{x\to
x_{0}}\frac{|u(x)-u(x_{0})+\|Du\|_{L^{\infty}(\Omega_{0})}|x-x_{0}||}{|x-x_{0}|}=0.$

Given any $\epsilon>0$, there is $r_{\epsilon}<1$ such that

$\displaystyle\frac{u(x)-u(x_{0})}{|x-x_{0}|}\geq\|Du\|_{L^{\infty}(\Omega_{0})}-\epsilon\quad\forall
x\in B(x_{0},r_{\epsilon})\backslash\\{x_{0}\\}\Subset\Omega_{0}.$ (8.1)

For each $x\in B(x_{0},r_{\epsilon})\backslash\\{x_{0}\\}$, by Lemma 4.1 we
get

$|Du(x)|\geq\frac{u(x)-u(x+t\nu_{x})}{t}\quad\forall 0<t<r_{x,\nu_{x}},$

where $\nu_{x}=-Du(x)/|Du(x)|$ and $u(x+r_{x,\nu_{x}}\nu_{x})=0$. Observe
that, given any $t\in(0,\min\\{r_{\epsilon},r_{x,\nu_{x}}\\})$, via (8.1) we
obtain

$\displaystyle\liminf_{x\to x_{0}}|Du(x)|$ $\displaystyle\geq\liminf_{x\to
x_{0}}\frac{u(x)-u(x+t\nu_{x})}{t}$
$\displaystyle\geq\frac{u(x_{0})-\sup_{|z|=1}u(x_{0}+tz)}{t}$
$\displaystyle\geq\|Du\|_{L^{\infty}(\Omega_{0})}-\epsilon.$

By the arbitrariness of $\epsilon>0$, we have $\liminf_{x\to
x_{0}}|Du(x)|\geq\|Du\|_{L^{\infty}(\Omega_{0})}$ as desired. ∎

We are ready to prove Theorem 1.4.

###### Proof of Theorem 1.4.

If $\Omega=B(x_{0},r)\setminus\\{x_{0}\\}$ for some $r>0$, then the
$\infty$-harmonic potential $u(x)=1-|x-x_{0}|/r$. Obviously, $u\in
C^{2}(\Omega)$ and also $u$ is concave. Thus (ii) and (iii) follow from (i).

Now we prove (ii) $\Rightarrow$ (i). Assume $u\in C^{2}(\Omega)$. Obviously,
$B(x_{0},{\rm dist}(x_{0},\partial\Omega_{0}))\subset\Omega_{0}$. The proof of
(i) is reduced to proving $\Omega_{0}\subset B(x_{0},{\rm
dist}(x_{0},\partial\Omega_{0})).$ Given any $x\in\Omega$, we show the desired
$|x-x_{0}|<{\rm dist}(x_{0},\partial\Omega_{0})$ as below.

Using Corollary 1.2, there exists a curve ${\gamma}_{x}\in
C^{0}([0,T_{x}];\Omega_{0})\cap C^{1}([0,T_{x});\Omega)$ for some
$0<T_{x}<\infty$ such that

$\displaystyle\frac{d{\gamma}_{x}(t)}{dt}=Du({\gamma}_{x}(t))\quad\forall
t\in[0,T_{x});\quad{\gamma}_{x}(0)=x,\quad{\gamma}_{x}(T_{x})=x_{0}.$ (8.2)

Note that $u\in C^{2}(\Omega)$, by $D^{2}uDu\cdot Du=0$ in $\Omega$ one has

$\frac{d}{dt}|Du|^{2}({\gamma}_{x}(t))=D|Du|^{2}({\gamma}_{x}(t))\cdot
Du({\gamma}_{x}(t))=0$

and hence

$|Du({\gamma}_{x}(0)|=|Du({\gamma}_{x}(t))|\quad\forall t\in[0,T_{x}).$

Applying Lemma 8.1 one gets

$|Du(x)|=\left|\frac{d{\gamma}_{x}(t)}{dt}\right|=|Du({\gamma}_{x}(t))|=\lim_{s\to
T_{x}^{-}}|Du({\gamma}_{x}(s))|=\frac{1}{{\rm
dist}(x_{0},\partial\Omega_{0})}\quad\forall t\in[0,T_{x}).$

Thanks to this, for any $t\in(0,T_{x})$ we obtain

$\displaystyle
u({\gamma}_{s}(t))-u({\gamma}_{x}(0))=\int^{t}_{0}Du({\gamma}_{x}(t))\cdot\frac{d{\gamma}_{x}(t)}{dt}\,dt=\int^{t}_{0}|Du({\gamma}_{x}(t))|\left|\frac{d{\gamma}_{x}(t)}{dt}\right|\,dt,$

by $1>u(x_{0})-u(x)\geq u({\gamma}_{s}(t))-u(x)>0$ we conclude that

$\displaystyle 1>u(x_{0})-u(x)=\frac{1}{{\rm
dist}(x_{0},\partial\Omega_{0})}\int^{t}_{0}\left|\frac{d{\gamma}_{x}(t)}{dt}\right|\,dt\geq\frac{|{\gamma}_{x}(t)-{\gamma}_{x}(0)|}{{\rm
dist}(x_{0},\partial\Omega_{0})}\quad\forall t\in(0,T_{x}).$

By ${\gamma}_{x}\in C^{0}([0,T_{x}])$ and ${\gamma}(T_{x})=x_{0}$, letting
$t\to T_{x}^{-}$ we have

$1>\frac{|x-x_{0}|}{{\rm dist}(x_{0},\partial\Omega_{0})}$

as desired.

Finally we prove (iii) $\Rightarrow$ (i). Assume that $u$ is concave on
$\Omega$. Similarly, we only need to show $\Omega_{0}\subset B(x_{0},{\rm
dist}(x_{0},\partial\Omega_{0})).$ We claim that $|Du|$ is a constant. If so,
then by Lemma 8.1
$|Du|=\frac{1}{{\mathop{\mathrm{missing}}{\,dist\,}}(x_{0},\partial\Omega_{0})}$.
By Corollary 1.2, for any $x\in\Omega$ there is a curve ${\gamma}_{x}\in
C^{0}([0,T_{x}];\Omega_{0})\cap C^{1}([0,T_{x});\Omega)$ for some
$0<T_{x}<\infty$ so that (8.2) holds. Via an argument similarly to above, for
all $x\in\Omega_{0}$ we have
$|x-x_{0}|<{\mathop{\mathrm{missing}}{\,dist\,}}(x_{0},\partial\Omega)$ as
desired.

To prove the claim that $|Du|$ is a constant, since $|Du|^{2}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ we only need to show
that $D|Du|^{2}=0$ almost everywhere. The concavity of $u$ implies that $u$ is
twice differentiable almost everywhere on $\Omega$; see [25, Theorem 2.3.1].
Denote by $D^{2}u$ as its Hessian matrix. Note that $D^{2}u$ is symmetric
almost everywhere. Thanks to Theorem 1.2, $D|Du|^{2}\cdot Du=0$ almost
everywhere, and therefore $D^{2}uDu\cdot Du=0$ almost everywhere. At each such
point one has $D|Du|^{2}=\frac{1}{2}D^{2}uDu$. So it further suffices to show
that $D^{2}uDu=0$ almost everywhere.

Given any point $x$, where $D^{2}u$ is symmetric and $D^{2}uDu\cdot Du=0$, for
any $\xi\in{{{\mathbb{R}}}^{n}}\setminus\\{0\\}$ one has

$u(x+t\xi)=u(x)+tDu(x)\cdot\xi+\frac{t^{2}}{2}\xi^{T}D^{2}u(x)\xi+o(t^{2})\quad\mbox{as
$t\to 0$}.$

Since the concavity of $u$ implies that

$u(y)\leq u(x)+Du(x)\cdot(y-x)\quad\forall y,x\in\Omega.$

By setting $y=x+t\xi$ for sufficiently small $t$, and letting $t\to 0$ we
conclude that

$\displaystyle D^{2}u(x)\xi\cdot\xi\leq\lim_{t\to
0}\frac{o(t^{2})}{t^{2}}=0\quad\forall\xi\in{\mathbb{R}}^{n}.$

Taking $\xi=\frac{1}{\epsilon}Du(x)+\epsilon D^{2}u(x)Du(x)$ with
$\epsilon>0$, at $x$ we get

$\displaystyle\frac{1}{\epsilon^{2}}D^{2}uDu\cdot
Du+2|D^{2}uDu|^{2}+\epsilon^{2}D^{2}uD^{2}uDu\cdot D^{2}uDu\leq 0.$

Since $D^{2}uDu\cdot Du=0$ at point $x$ we get

$\displaystyle 2|D^{2}uDu|^{2}+\epsilon^{2}D^{2}uD^{2}uDu\cdot D^{2}uDu\leq
0.$

Letting $\epsilon\to 0$, at $x$ we have $|D^{2}uDu|^{2}=0$ as desired.

∎

##  9 Proof of Theorem 1.5

In this section we prove Theorem 1.5 by using Theorem 1.3 and a fundamental
inequality in Lemma 6.1.

Let us start by proving the following upper bound of $D^{2}u_{p}$ for the
$p$-harmonic potential $u_{p}$ in $\Omega$.

######  Lemma 9.1.

Let $p\in(4,\infty)$. For all $B(z,r)\Subset\Omega$ we have

$\displaystyle\int_{B(z,r)}|D^{2}u_{p}|\xi^{2}\,dx\leq 2\int_{B(z,r)}-\Delta
u_{p}\xi^{2}\,dx+2\int_{B(z,r)}|D|Du_{p}||\xi^{2}\,dx\quad\forall\xi\in
C^{\infty}_{c}(B(z,r)).$

###### Proof.

Since $Du_{p}\neq 0$ on $\Omega$, recalling that Lemma 6.1 give us

$|D^{2}u_{p}|^{2}-(\Delta u_{p})^{2}\leq
2[|D|Du_{p}||^{2}-\Delta_{\infty}^{N}u_{p}\Delta u_{p}]\quad{\rm
in}\quad\Omega.$

Using equation $-\Delta_{\infty}^{N}u_{p}=\frac{1}{p-2}\Delta u_{p}$ in
$\Omega$, we have

$|D^{2}u_{p}|^{2}\leq[1+\frac{2}{p-2}](\Delta u_{p})^{2}+2|D|Du_{p}||^{2}\leq
2[(\Delta u_{p})^{2}+|D|Du_{p}||^{2}]\quad{\rm in}\quad B(z,r)\Subset\Omega,$

which immediately implies

$|D^{2}u_{p}|\leq 2|\Delta u_{p}|+2|D|Du_{p}||\quad{\rm in}\quad
B(z,r),\quad\forall p\in(4,\infty).$

Multiplying both sides by a test function $\xi^{2}$ with $\xi\in
C^{\infty}_{c}(B(z,r))$, and integrating over on $B(z,r)$, we get

$\displaystyle\int_{B(z,r)}|D^{2}u_{p}|\xi^{2}\,dx\leq 2\int_{B(z,r)}|\Delta
u_{p}|\xi^{2}\,dx+2\int_{B(z,r)}|D|Du_{p}||\xi^{2}\,dx.$

Since $-\Delta u_{p}\geq 0$ on $\Omega$ (see [41]), we obtain the desired
inequality. ∎

By Lemma 7.1, we have the following upper bound uniformly in all $p>4$.

######  Lemma 9.2.

For $p\in(4,\infty)$, there exists a universal constant $C$ such that

$\displaystyle\int_{B(z,r/2)}|D^{2}u_{p}|\,dx$ $\displaystyle\leq
C\frac{1}{r}\int_{B(z,r)}|Du_{p}|\,dx$
$\displaystyle\quad\quad+C\frac{1}{r}\max_{B(z,r)}|Du_{p}|^{2}\left(\int_{B(z,r)}|Du_{p}|^{-2}\,dx\right)^{\frac{1}{2}}\quad\mbox{
whenever $B(z,r)\Subset\Omega$.}$

###### Proof.

Thanks to Lemma 9.1 and Lemma 7.1, we only need to bound $\int_{B(z,r)}-\Delta
u_{p}\xi^{2}\,dx$. Via integration by parts we obtain

$\displaystyle 2\int_{B(z,r)}-\Delta u_{p}\xi^{2}\,dx$
$\displaystyle=-2\int_{B(z,r)}\Delta
u_{p}\xi^{2}\,dx=4\int_{B(z,r)}Du_{p}\cdot D\xi\xi\,dx\quad\forall\xi\in
C^{\infty}_{c}(B(z,r)).$

By choosing a suitable cut-off function $\xi$, one has

$2\int_{B(z,r)}-\Delta
u_{p}\xi^{2}\,dx\leq\frac{1}{r}\int_{B(z,r)}|Du_{p}|\,dx.$

∎

Letting $p\to\infty$ we conclude the following from Lemma 9.2. We write
$\|\mu\|$ as the variation measure of a signed Radon measure $\mu$. If $\mu$
is nonnegative, then $\|\mu\|=\mu$.

######  Lemma 9.3.

The distributional second order derivatives $\mathcal{D}^{2}u$ are Radon
measures satisfying

$\displaystyle\int_{\Omega}\langle\mathcal{D}^{2}uDu,Du\rangle\xi\,dx=0\quad\forall\xi\in
C^{\infty}_{c}(\Omega),$ (9.1)

and also $D^{2}u_{p}\to\mathcal{D}^{2}u$ weakly in the sense of measure.

It holds that $Du\in BV_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$, that
is, the distributional derivatives $\mathcal{D}[Du]$ of $Du$ are Radon
measures, and that $\mathcal{D}[Du]=\mathcal{D}^{2}u$.

Moreover, there exists a universal constant $C$ such that

$\displaystyle\|\mathcal{D}^{2}u\|(B(z,r/2))$ $\displaystyle\leq
C\frac{1}{r}\int_{B(z,r)}|Du|\,dx$
$\displaystyle\quad\quad+C\frac{1}{r}\max_{B(z,r)}|Du|^{2}\left(\int_{B(z,r)}|Du|^{-2}\,dx\right)^{\frac{1}{2}}\quad\mbox{
whenever $B(z,r)\Subset\Omega$,}$ (9.2)

and also

$\displaystyle\|\mathcal{D}^{2}u\|(B(z,r/2))\leq 2[-\Delta
u](B(z,r))+2\int_{B(z,r)}|D|Du||\,dx\quad\mbox{ whenever
$B(z,r)\Subset\Omega$,}$ (9.3)

where $-\Delta u$ is a nonnegative Radon measure.

###### Proof.

Applying Lemma 9.2 and Lemma 4.4, we know that $D^{2}u_{p}\in
L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ uniformly in all $p>4$.
Via the compactness of the space $\mathcal{M}(K)$ of Radon measures in any
compact subset $K\Subset\Omega$, for all $1\leq i,j\leq n$ we can find a
locally finite Radon measure $\mu_{ij}$ so that
$\frac{\partial^{2}u_{p}}{\partial x_{i}\partial x_{j}}$ weakly converges to
$\mu_{ij}$ in each $K\Subset\Omega$ in the sense of measure, that is,

$\int_{\Omega}\frac{\partial^{2}u_{p}}{\partial x_{i}\partial
x_{j}}\phi\,dx\to\mu_{ij}(\phi)\quad\forall\phi\in C^{\infty}_{c}(\Omega).$

On the other hand, since $u_{p}\to u$ in $C^{0}(\Omega)$, we know that

$\int_{\Omega}\frac{\partial^{2}u_{p}}{\partial x_{i}\partial
x_{j}}\phi\,dx=\int_{\Omega}u_{p}\frac{\partial^{2}\phi}{\partial
x_{i}\partial x_{j}},dx\to\int_{\Omega}u\frac{\partial^{2}\phi}{\partial
x_{i}\partial x_{j}}dx\quad\forall\phi\in C^{\infty}_{c}(\Omega).$

That is, $\mu_{ij}$ coincides with the distributional second order partial
derivatives $\mathcal{D}_{ij}u$ of $u$. Note that
$\mathcal{D}_{ji}u=\mathcal{D}_{ij}u$, we have $\mu_{ij}=\mu_{ji}$.

Since

$\int_{\Omega}\mathcal{D}_{j}u_{x_{i}}\phi\,dx=-\int_{\Omega}u_{x_{i}}\phi_{x_{j}}\,dx=\int_{\Omega}u\phi_{x_{j}x_{i}}\,dx=\int_{\Omega}\mathcal{D}_{x_{i}x_{j}}u\phi\,dx,$

we know that the distributional derivative $\mathcal{D}_{j}u_{x_{i}}=\mu_{ij}$
is also a Radon measure, that is, $u_{x_{i}}\in
BV_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$.

Recall that $Du_{p}\to Du$ in $C^{0}(\Omega)$ in Theorem 1.1 and $-\Delta
u_{p}\geq 0$ in $\Omega$, then passing to the limit $p\to\infty$ in Lemma 9.1
and Lemma 9.2 we get (9.3) and (9.3) as desired. In particular, we have that
$-\Delta u$ is a nonnegative Radon measure. Similarly, using equation
$D^{2}u_{p}Du_{p}\cdot Du_{p}=-\frac{1}{p-2}\Delta u_{p}$ we conclude (9.1)
follows by letting $p\to\infty$.

∎

We now come to prove Theorem 1.5.

###### Proof of Theorem 1.5.

This follows from Lemma 9.3. ∎

##  10 Proof of Theorem 1.6

In this section we prove Theorem 1.6. We assume that dimension $n=2$.

We start by showing the following.

######  Lemma 10.1.

For all $\alpha\in{\mathbb{R}}$, $|Du|^{\alpha}$ is differentiable almost
everywhere in $\Omega$. For $i=1,2$, $u_{x_{i}}$ and $u^{2}_{x_{i}}$ are
differentiable almost everywhere in $\Omega$.

The following can be concluded from Rademacher’s theorem [54]; see also
Stepanov in [55].

######  Lemma 10.2.

Let $U\subset{\mathbb{R}}^{n}$ be any domain. A function $g:U\to{\mathbb{R}}$
is differentiable almost everywhere in $U$ if and only if
${\mathop{\mathrm{missing}}{\,Lip}}\,g(x)<\infty$ for almost all $x\in U.$

###### Proof of Lemma 10.1.

We first show that $|Du|^{\alpha}$ is differentiable almost everywhere in
$\Omega$ for all $\alpha\in{\mathbb{R}}$. Recall that Theorem 1.1 gives us
$u\in C^{1}(\Omega)$ and $|Du|\neq 0$ in $\Omega$. It suffices to prove the
almost everywhere differentiability of $|Du|^{2}$. Thanks to Lemma 10.2, we
only need to prove that
${\mathop{\mathrm{missing}}{\,Lip}}(|Du|^{2})(z)<\infty$ for almost all
$z\in\Omega$. To this end, using (B.5) in Lemma B.3, we have

$\displaystyle{\mathop{\mathrm{missing}}{\,Lip}}(|Du|^{2})(z)$
$\displaystyle=\lim_{r\to 0}\sup_{y\in
B(z,r)}\frac{|Du(y)|^{2}-|Du(z)|^{2}}{r}$ $\displaystyle\leq\limsup_{r\to
0}\mathop{\mathrm{missing}}{\,osc\,}_{B(z,r)}\frac{|Du|^{2}}{r}$
$\displaystyle\leq\limsup_{r\to 0}\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}|D|Du|^{2}|^{2}\,dx\right)^{1/2},$

which is finite whenever $z$ is a Lebesgue point of $|D|Du|^{2}|^{2}$. Since
$|D|Du|^{2}|^{2}\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$, its
Lebesgue points is dense in $\Omega$. Thus
${\mathop{\mathrm{missing}}{\,Lip}}(|Du|^{2})(z)<\infty$ for almost all
$z\in\Omega$ as desired.

Next, we show the almost everywhere differentiability of $u_{x_{i}}$, which
gives the almost everywhere differentiability of $u^{2}_{x_{i}}$ in an obvious
way. Similarly to above, it suffices to prove that
${\mathop{\mathrm{missing}}{\,Lip}}(u_{x_{i}})(z)<\infty$ for almost all
$z\in\Omega$. By (B.6) in Lemma B.3 and Lemma 9.3, we have

$\displaystyle{\mathop{\mathrm{missing}}{\,Lip}}(u_{x_{i}})(z)$
$\displaystyle=\limsup_{r\to 0}\sup_{y\in
B(z,r)}\frac{u_{x_{i}}(y)-u_{x_{i}}(z)}{r}$ $\displaystyle\leq\limsup_{r\to
0}\frac{\mathop{\mathrm{missing}}{\,osc\,}_{B(z,r)}u_{x_{i}}}{r}$
$\displaystyle\leq\limsup_{r\to 0}\frac{\|D^{2}u\|(B(z,r/2))}{r^{2}}$
$\displaystyle\leq C\limsup_{r\to 0}\frac{\|-\Delta
u\|(B(z,r))}{r^{2}}+C\limsup_{r\to 0}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}|D|Du||\,dx.$
(10.1)

Thanks to $|D|Du||\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$, at
its Lebesgue points and hence almost all points in $\Omega$, one has

$\displaystyle\limsup_{r\to 0}\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}|D|Du||\,dx<\infty\quad\mbox{for
almost all $z\in\Omega$.}$ (10.2)

Thanks to (10) and (10.2), in order to show
${\mathop{\mathrm{missing}}{\,Lip}}(u_{x_{i}})(z)<\infty$ for almost all
$z\in\Omega$, it suffices to prove that

$\displaystyle\limsup_{r\to 0}\frac{\|-\Delta u\|(B(z,r))}{r^{2}}<\infty\
\mbox{ for almost all $z\in\Omega$. }$ (10.3)

Below we prove (10.3). Recall that $Du\in
BV_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ and the distributional
$[-\Delta u]$ is a nonnegative Radon measure. We therefore write $[-\Delta u]$
as the absolutely continuous part $[-\Delta u]_{ac}$ and the singular part
$[-\Delta u]_{s}$ with respect to the Lebesgue measure, that is,

$[-\Delta u]=[-\Delta u]=[-\Delta u]_{ac}+[-\Delta u]_{s};$

see [29, Chapter 6]. By [29, Theorem 6.1] we have

$\displaystyle\lim_{r\to 0}\frac{[-\Delta
u]_{s}(B(z,r))}{r^{2}}=0\quad\mbox{for almost all $z\in\Omega$}.$ (10.4)

Therefore

$\displaystyle\lim_{r\to 0}\frac{[-\Delta u](B(z,r))}{r^{2}}=\lim_{r\to
0}\frac{[-\Delta u]_{ac}(B(z,r))}{r^{2}}\quad\mbox{for almost all
$z\in\Omega$}.$

Denote by $g\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ the Radon-
Nikodym derivative of $[-\Delta u]_{ac}$, that $[-\Delta u]_{ac}=g\,dx$. At
any Lebesgue points $z$, we have

$\frac{[-\Delta u]_{ac}(B(z,r))}{r^{2}}=\lim_{r\to
0}\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}g(y)\,dy=g(z).$

Thanks to $g\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ and the
density of Lebesgue points of $g$ in $\Omega$ we attain

$\displaystyle\lim_{r\to 0}\frac{[-\Delta
u]_{ac}(B(z,r))}{r^{2}}=g(z)<\infty\quad\mbox{for almost all $z\in\Omega$}.$

From this and (10.4) we conclude (10.3) as desired. ∎

######  Remark 10.3.

Since $|Du|^{2}$ is monotone and $|Du|^{2}\in
W^{1,2}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$, by Onninen [49,
Theorem 1.2], $|Du|^{2}$ is differentiable almost everywhere. Here we give a
direct proof via Lemma B.3.

Next, we show the following property for the absolutely continuous part of
Radon measure $\mathcal{D}^{2}u$.

######  Lemma 10.4.

For $1\leq i,j\leq 2$, the absolutely continuous part of the measure
$\mathcal{D}^{2}_{ji}u$ is given by $u_{x_{i}x_{j}}\,dx$, that is,

$[\mathcal{D}^{2}u]_{ac}=[\mathcal{D}(Du)]_{ac}=D^{2}u\,dx.$

In particular, $D^{2}u\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$
is symmetric almost everywhere, that is, $u_{x_{1}x_{2}}=u_{x_{2}x_{1}}$
almost everywhere in $\Omega$.

###### Proof.

Since $u_{x_{i}}\in BV_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$, the
measure $\mathcal{D}_{j}u_{x_{i}}$ is decomposed as the absolutely continuous
part $[\mathcal{D}_{j}u_{x_{i}}]_{ac}$ and the singular part
$[\mathcal{D}_{j}u_{x_{i}}]_{s}$ with respect to the Lebesgue measure. Denote
by $g_{ij}\in L^{1}_{\mathop{\mathrm{missing}}{\,loc\,}}(\Omega)$ the Radon-
Nikodym derivative of $[\mathcal{D}_{j}u_{x_{i}}]_{ac}$ with respect to the
Lesbesgue measure, that is, $[\mathcal{D}_{j}u_{x_{i}}]_{ac}=g_{ij}\,dx$.
Since
$\mathcal{D}_{1}u_{x_{2}}=\mathcal{D}_{12}u=\mathcal{D}_{21}u=\mathcal{D}_{2}u_{x_{1}}$,
we know that $g_{12}=g_{21}$ almost everywhere in $\Omega$. Let $z$ be any
Lebesgue point of $\\{g_{ij}\\}_{j=1,2}$ so that $g_{12}(z)=g_{21}(z)$ and
$Du$ is differentiable at $z$. Note that the set of all such $z$ is dense in
$\Omega$. Applying [29, Theorem 6.1] we have

$\displaystyle\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}|u_{x_{i}}(x)-u_{x_{i}}(z)-\sum_{j=1}^{2}g_{ij}(z)\cdot(x_{j}-z_{j})|^{2}\,dx\right)^{\frac{1}{2}}=o(r).$
(10.5)

Since $u_{x_{i}}$ is differentiable at $z$, we have

$\displaystyle\left|u_{x_{i}}(x)-u_{x_{i}}(z)-\sum_{j=1}^{2}u_{x_{i}x_{j}}(z)(x_{j}-z_{j})\right|=o(r).$
(10.6)

Thus

$\displaystyle\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to21.63203pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to21.63203pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to15.9276pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B(z,r)}\sum_{i=1}^{2}|\sum_{j=1}^{2}[g_{ij}(z)-u_{x_{i}x_{j}}(z)](x_{j}-z_{j})|^{2}\,dx\right)^{\frac{1}{2}}=o(r).$

This implies that $g_{ij}(z)=u_{x_{i}x_{j}}(z)$ for all possible $i,j$. ∎

It follows from Lemmas 10.1 and 10.4 that $u$ is twice differentiable almost
everywhere in $\Omega$.

######  Lemma 10.5.

It holds that $u$ is twice differentiable almost everywhere in $\Omega$, that
is, for almost all $z\in\Omega$,

$\lim_{r\to 0}\sup_{x\in
B(z,r)}\frac{\left|u(x)-u(z)-Du(z)\cdot(x-z)-\frac{1}{2}(x-z)^{T}\cdot
D^{2}u(x)(x-z)\right|}{r^{2}}=0.$

###### Proof.

Let $z\in\Omega$ be any point where $Du$ is differentiable at $z$. Without
loss of generality, we assume that $z=0$. Writing

$u(x)-u(0)=\int^{1}_{0}x\cdot Du(tx)\,dt,$

we have

$\displaystyle u(x)-u(0)+Du(0)\cdot x=\int^{1}_{0}x\cdot[Du(tx)-Du(0)]\,dt.$
(10.7)

Since $Du$ is differentiable at $0$, for each $\epsilon\in(0,1)$ there is
$r_{\epsilon}>0$ such that for $r<r_{\epsilon}$ one has

$\displaystyle|Du(y)-Du(0)-D^{2}u(0)y|\leq\epsilon r\quad\forall y\in B(0,r).$
(10.8)

If $x\in B(0,r)$ and $0<t<1$, applying (10.8) to $y=tx$ we have

$|x\cdot Du(tx)-x\cdot Du(0)-x\cdot D^{2}u(0)tx]|\leq\epsilon r^{2}.$

Thus

$\left|\int_{0}^{1}[x\cdot Du(tx)-x\cdot Du(0)-x\cdot
D^{2}u(0)tx]\,dt\right|\leq\epsilon r^{2}$

and hence by (10.7) we get

$\left|u(x)-u(0)+Du(0)\cdot x-\frac{1}{2}x^{T}\cdot
D^{2}u(0)x\right|\leq\epsilon r^{2}$

as desired. ∎

###### Proof of Theorem 1.6.

Thanks to Lemma 10.4 and Lemma 10.5, we only need to prove that
$-D^{2}uDu\cdot Du=0$, (1.5) and (1.6) hold almost everywhere in $\Omega$.

Since $|Du|^{2},u^{2}_{x_{1}},u^{2}_{x_{2}}$ are differentiable almost
everywhere, we know that

$\displaystyle
D|Du|^{2}=D[u_{x_{1}}^{2}+u_{x_{2}}^{2}]=2(u_{x_{1}x_{1}}u_{x_{1}}+u_{x_{2}x_{1}}u_{x_{2}},u_{x_{1}x_{2}}u_{x_{1}}+u_{x_{2}x_{2}}u_{x_{2}})=2D^{2}uDu$

almost everywhere. Recall that $|Du|^{2}\in
W^{1,2}_{\mathop{\mathrm{missing}}{\,loc\,}}$ and $D|Du|^{2}\cdot Du=0$ almost
everywhere, we have $-D^{2}uDu\cdot Du=0$ almost everywhere.

Next we show (1.5) and (1.6) at any point $x$, where $u$ is twice
differentiable at $x$ and $-D^{2}uDu\cdot Du=0$ at $x$. Without loss of
generality we write $x=0$. By Taylor’s expansion, we have

$u(z)=u(0)+Du(0)\cdot z+\frac{1}{2}D^{2}u(0)z\cdot z+o(|z|^{2})\quad\mbox{as
$|z|\to 0$.}$

Since $Du\neq 0$ on $\Omega$, choosing $z=\pm hDu(0)$, we get

$u(\pm hDu(0))=u(0)\pm h|Du(0)|+\frac{1}{2}h^{2}D^{2}u(0)Du(0)\cdot
Du(0)+o(h^{2})\quad\mbox{as $|h|\to 0$ },$

and hence

$u(0)=\frac{1}{2}\left[u\left(hDu(0)\right)+u\left(-hDu(0)\right)\right]+o(h^{2})\quad\mbox{as
$|h|\to 0$ },$

which gives (1.5) at $x=0$ as desired.

To get (1.6) at $x=0$, choose $|x^{\pm}_{\epsilon}|=\epsilon$ so that

$\max_{\overline{B(0,\epsilon)}}u=u(x^{+}_{\epsilon})\ \mbox{and}\
\min_{\overline{B(0,\epsilon)}}u=u(x^{-}_{\epsilon}).$

Recalling that $u\in C^{1}(\Omega)$ and $Du\neq 0$ on $\Omega$, and noting
that at $x^{\pm}_{\epsilon}$ the tangential derivatives of $u$ along $\partial
B_{\epsilon}$ are zero, we have

$\displaystyle\frac{Du(x^{\pm}_{\epsilon})}{|Du(x^{\pm}_{\epsilon})|}=\pm\frac{x^{\pm}_{\epsilon}}{|x^{\pm}_{\epsilon}|}.$
(10.9)

Observe that $\frac{Du}{|Du|}$ is differentiable at $0$ due to Lemma 10.1, we
have

$\displaystyle\frac{Du(x^{\pm}_{\epsilon})}{|Du(x^{\pm}_{\epsilon})|}$
$\displaystyle=\frac{Du(0)}{|Du(0)|}+O(\epsilon).$ (10.10)

On the other hand, by Taylor’s expansion, one has

$\displaystyle\max_{\overline{B(0,\epsilon)}}u+\min_{\overline{B(0,\epsilon)}}u$
$\displaystyle\leq
u(x^{+}_{\epsilon})+u(-x^{+}_{\epsilon})=2u(0)+D^{2}u(0)x^{+}_{\epsilon}\cdot
x^{+}_{\epsilon}+o(\epsilon^{2})$ (10.11)

and

$\displaystyle\max_{\overline{B(0,\epsilon)}}u+\min_{\overline{B(0,\epsilon)}}u$
$\displaystyle\geq
u(x^{-}_{\epsilon})+u(-x^{-}_{\epsilon})=2u(0)+D^{2}u(0)x^{-}_{\epsilon}\cdot
x^{-}_{\epsilon}+o(\epsilon^{2}).$ (10.12)

Using (10.9), (10.10) and $D^{2}uDu\cdot Du=0$ at point $0$ we have

$\displaystyle D^{2}u(0)x^{\pm}_{\epsilon}\cdot x^{\pm}_{\epsilon}$
$\displaystyle=\epsilon^{2}D^{2}u(0)\frac{Du(x^{\pm}_{\epsilon})}{|Du(x^{\pm}_{\epsilon})|}\cdot\frac{Du(x^{\pm}_{\epsilon})}{|Du(x^{\pm}_{\epsilon})|}$
$\displaystyle=\epsilon^{2}D^{2}u(0)[\frac{Du(0)}{|Du(0)|}+O(\epsilon)]\cdot[\frac{Du(0)}{|Du(0)|}+O(\epsilon)]$
$\displaystyle=\epsilon^{2}D^{2}u(0)\frac{Du(0)}{|Du(0)|}\cdot\frac{Du(0)}{|Du(0)|}+O(\epsilon^{3})$
$\displaystyle=o(\epsilon^{2}).$

From (10.11) and (10.12), it follows that

$2u(0)+o(\epsilon^{2})\leq\max_{\overline{B(0,\epsilon)}}u+\min_{\overline{B(0,\epsilon)}}u\leq
2u(0)+o(\epsilon^{2}),$

which gives (1.5) at $x=0$ as desired. ∎

##  Appendix A Proof of Corollary 1.2

Theorem 1.1 allows us to borrow some idea from Lindgren-Lindqvist [44] to
prove Corollary 1.2. We give the details for reader’s convenience.

###### Proof of Corollary 1.2.

Fix any $x\in\Omega=\Omega_{0}\backslash\overline{\Omega}_{1}$. We split the
proof into 2 steps.

Step 1. Given any $p>2$, since $u_{p}\in C^{\infty}(\Omega)$, by [34,
Corollary 2.3] there exists a unique solution ${\gamma}^{p}_{x}\in
C^{1}([0,T_{x}^{p}),\Omega)$ for some $T^{p}_{x}\in(0,\infty]$ to the problem

$\displaystyle\frac{d{\gamma}^{p}_{x}(t)}{dt}$
$\displaystyle=Du_{p}({\gamma}^{p}_{x}(t))\quad\forall t\in[0,T_{x}^{p});\
{\gamma}^{p}_{x}(0)=x,$

where $[0,T_{x}^{p})$ is the maximal time interval. Set

$\Phi_{x}^{p}(t):=u_{p}({\gamma}^{p}_{x}(t))\quad\forall t\in[0,T_{x}^{p}).$

Observe that

$\frac{d\Phi_{x}^{p}(t)}{dt}=|Du_{p}({\gamma}_{x}^{p}(t))|^{2}\geq
0\quad\forall t\in[0,T_{x}^{p}).$

Thus, $\Phi_{x}^{p}$ is nondecreasing in $[0,T_{x}^{p})$. Moreover, we compute
that

$\displaystyle\frac{d^{2}\Phi_{x}^{p}(t)}{dt^{2}}$
$\displaystyle=\frac{d}{dt}|Du_{p}({\gamma}_{x}^{p}(t))|^{2}=2D^{2}u_{p}({\gamma}_{x}^{p}(t))Du_{p}({\gamma}_{x}^{p}(t))\cdot\frac{d{\gamma}^{p}_{x}(t)}{dt}=2\Delta_{\infty}u_{p}({\gamma}_{x}^{p}(t)).$

Since

$\mbox{$\Delta_{p}u_{p}=|Du_{p}|^{p-2}(\Delta
u_{p}+(p-2)\Delta^{N}_{\infty}u_{p})=0$, $Du_{p}\neq 0$ and $-\Delta u_{p}\geq
0$ in $\Omega$},$

it follows that

$\displaystyle\frac{d^{2}\Phi_{x}^{p}(t)}{dt^{2}}$
$\displaystyle=-\frac{2}{p-2}|Du_{p}({\gamma}^{p}_{x}(t))|^{2}\Delta
u_{p}({\gamma}^{p}_{x}(t))\geq 0,$

which means that $\Phi_{x}^{p}$ is convex on $[0,T_{x}^{p})$ and
$|Du_{p}\circ{\gamma}^{p}_{x}|$ is nondecreasing in $[0,T_{x}^{p})$.

Now using the fact that $|Du_{p}\circ{\gamma}^{p}_{x}|$ is nondecreasing in
$[0,T_{x}^{p})$, via (4.10) in Lemma 4.2 we conclude that

$|Du_{p}({\gamma}_{x}^{p}(t))|\geq|Du_{p}({\gamma}_{x}^{p}(0))|=|Du_{p}(x)|\geq\frac{u_{p}(x)}{{\rm
diam}(\Omega_{0})}\quad\forall t\in[0,T_{x}^{p}).$

Since $u_{p}\to u$ in $C^{0}(\overline{\Omega})$ and $u(x)>0$, there exists a
fixed constant $p_{x}>2$ such that

$\frac{d\Phi_{x}^{p}(t)}{dt}\geq\left[\frac{1}{2}\frac{u(x)}{{\rm
diam}(\Omega_{0})}\right]^{2}>0\quad\forall t\in[0,T_{x}^{p})\quad\forall
p>p_{x}.$

This leads to

$\displaystyle
1\geq\Phi_{x}^{p}(t)-\Phi_{x}^{p}(0)=\int^{t}_{0}\frac{d\Phi_{x}^{p}(s)}{ds}\,ds\geq
t\left[\frac{1}{2}\frac{u(x)}{{\rm diam}(\Omega_{0})}\right]^{2}\quad\forall
0<t<T_{x}^{p},\quad\forall p>p_{x},$ (A.1)

and hence

$\displaystyle T_{x}^{p}\leq\left[\frac{1}{2}\frac{{\rm
diam}(\Omega_{0})}{u(x)}\right]^{2}<\infty\quad\forall p>p_{x}.$ (A.2)

Next we extend ${\gamma}_{x}^{p}$ to $[0,T_{x}^{p}]$ by setting
${\gamma}_{x}^{p}(T_{x}^{p})=\lim_{t\to{T_{x}^{p}}^{-}}{\gamma}_{x}^{p}(t).$
This comes from the fact that

$\displaystyle|{\gamma}_{x}^{p}(t)-{\gamma}_{x}^{p}(s)|$
$\displaystyle=\int_{s}^{t}|\frac{d{\gamma}^{p}_{x}(\delta)}{dt}|\,d\delta$
$\displaystyle=\int_{s}^{t}|Du_{p}({\gamma}^{p}_{x}(t))|\,d\delta$
$\displaystyle\leq(t-s)^{1/2}\left(\int_{s}^{t}|Du_{p}({\gamma}^{p}_{x}(\delta))|^{2}\,d\delta\right)^{1/2}$
$\displaystyle\leq(t-s)^{1/2}\left(\int_{s}^{t}\frac{d\Phi_{x}^{p}(\delta)}{dt}\,d\delta\right)^{1/2}$
$\displaystyle\leq(t-s)^{1/2}(\Phi_{x}^{p}(t)-\Phi_{x}^{p}(s))^{1/2}$
$\displaystyle\leq(t-s)^{1/2}\quad\forall 0\leq s<t<T_{x}^{p}.$ (A.3)

Note that ${\gamma}_{x}^{p}(T_{x}^{p})\in\partial\overline{\Omega}_{1}$ and
hence $u_{p}({\gamma}_{x}^{p}(T_{x}^{p}))=1$. Indeed, thanks to
$u_{p}({\gamma}_{x}^{p}(T_{x}^{p}))\geq u_{p}(x)$ we know that
${\gamma}_{x}^{p}(T_{x}^{p})\notin\partial\Omega_{0}$; since $[0,T_{x}^{p})$
is the maximal interval, ${\gamma}_{x}^{p}(T_{x}^{p})\notin\Omega$.

Step 2. By (A.2), $T_{x}^{p}$ is bounded uniformly in $p>p_{x}$, and hence, up
to some subsequence we may assume that $T_{x}^{p}\to T_{x}$ as $p\to\infty$.
By (Appendix A), ${\gamma}_{x}^{p}\in C^{1/2}([0,T_{x}^{p}])$ uniformly in
$p>p_{x}$. Thus $\\{{\gamma}_{x}^{p}\\}_{p>p_{x}}$ is uniformly bounded and
equal continuous in $[0,T_{x}]$, where if $T_{x}^{p}<T_{x}$ we let
${\gamma}_{x}^{p}(t)={\gamma}_{x}^{p}(T_{x}^{p})$ for $t\in[T_{x}^{p},T_{x}]$.
Therefore, we can find a curve ${\gamma}_{x}\in C^{0,1/2}([0,T_{x}])$ with
${\gamma}_{x}(0)=x$ and ${\gamma}_{x}(T_{x})\in\partial\Omega_{1}$, so that
${\gamma}_{x}^{p}\to{\gamma}_{x}$ in $C^{1/2}([0,T_{x}^{p}])$ as $p\to\infty$
(up to some subsequence).

Write $\Phi_{x}(t)=u({\gamma}_{x}(t))$ for all $t\in[0,T_{x}]$. Then
$\Phi_{x}^{p}\to\Phi_{x}$ in $C^{0}([0,T_{x}])$. Given any $s,t\in[0,T_{x})$,
due to $T_{x}^{p}\to T_{x}$, we know that $s,t\in[0,T_{x}^{p}]$ for all
sufficiently large $p$ (up to some subsequence). Thus

$\displaystyle\Phi_{x}(t)-\Phi_{x}(s)=\lim_{p\to\infty}[\Phi_{x}^{p}(t)-\Phi_{x}^{p}(s)]=\lim_{p\to\infty}\int_{s}^{t}\frac{d\Phi_{x}^{p}(\delta)}{d\delta}\,d\delta\geq(t-s)\left[\frac{u(x)}{2{\mathop{\mathrm{missing}}{\,diam\,}}(\Omega_{0})}\right]^{2},$

which implies that $\Phi_{x}$ is strictly increasing. Moreover, $\Phi_{x}$ is
convex on $[0,T_{x}]$ since

$\displaystyle\Phi_{x}(\lambda s+(1-\lambda)t)$
$\displaystyle=\lim_{p\to\infty}\Phi_{x}^{p}(\lambda s+(1-\lambda)t)$
$\displaystyle\leq\lim_{p\to\infty}[\lambda\Phi_{x}^{p}(s)+(1-\lambda)\Phi_{x}^{p}(t)]$
$\displaystyle=[\lambda\Phi_{x}(s)+(1-\lambda)\Phi_{x}(t)]\quad\forall
0<s<t<T_{x},\lambda\in(0,1),$

For any $0<t<T_{x}$, thanks to the strictly increasing property of $\Phi_{x}$,
we know that ${\gamma}_{x}([0,t])\subset\Omega$ and hence
${\gamma}_{x}^{p}([0,t])$ is contained in a neighborhood $U\Subset\Omega$ of
for all sufficiently large $p$ (up to some subsequence). Thus by Theorem 1.1
we have $Du_{p}\circ{\gamma}_{x}^{p}\to Du\circ{\gamma}_{x}$ in $C^{0}([0,t])$
as $p\to\infty$ (up to some subsequence). Since
$|Du_{p}\circ{\gamma}^{p}_{x}|$ is nondecreasing in $[0,T_{x}^{p})$, we know
that $|Du\circ{\gamma}_{x}|$ is also nondecreasing in $[0,T_{x})$. Moreover,
for any $0\leq s<t<T_{x}$, one has

$\displaystyle{\gamma}_{x}(t)-{\gamma}_{x}(s)$
$\displaystyle=\lim_{p\to\infty}[{\gamma}_{x}^{p}(t)-{\gamma}_{x}^{p}(s)]$
$\displaystyle=\lim_{p\to\infty}\int_{s}^{t}\frac{d{\gamma}_{x}^{p}(\delta)}{d\delta}\,d\delta$
$\displaystyle=\lim_{p\to\infty}\int_{s}^{t}Du_{p}({\gamma}_{x}^{p}(\delta))\,d\delta$
$\displaystyle=\int_{s}^{t}Du({\gamma}_{x}(\delta))\,d\delta,$

which implies that ${\gamma}_{x}\in C^{1}([0,T_{x}))$ with
$\frac{d{\gamma}_{x}(t)}{\,dt}=Du({\gamma}_{x}(t))$ for all $t\in(0,T_{x})$.
Since $Du\in L^{\infty}(\Omega)$, we also know that ${\gamma}_{x}\in
C^{0,1}([0,T_{x}])$. ∎

##  Appendix B Monotonicity in dimension $n=2$

In this section we assume $n=2$. For $p\in(2,\infty)$, denote by $u_{p}$ be
the $p$-harmonic potential in any given convex ring $\Omega$.

Note that $|Du_{p}|\neq 0$ on $\Omega$. The following monotonicity for $u_{p}$
is a direct consequence of the quasi-regular mapping due to Bojarski-Iwaniec
[13]. For the readers of convenience, we provide the proof in this Appendix.

######  Lemma B.1.

For all $B(z,r)\Subset\Omega$, we have

$\displaystyle\max_{\overline{B(z,r)}}|Du_{p}|=\max_{\partial
B(z,r)}|Du_{p}|,\quad\min_{\overline{B(z,r)}}|Du_{p}|=\min_{\partial
B(z,r)}|Du_{p}|$ (B.1)

and

$\displaystyle\max_{\overline{B(z,r)}}\frac{\partial u_{p}}{\partial
x_{i}}=\max_{\partial B(z,r)}\frac{\partial u_{p}}{\partial
x_{i}},\quad\min_{\overline{B(z,r)}}\frac{\partial u_{p}}{\partial
x_{i}}=\min_{\partial B(z,r)}\frac{\partial u_{p}}{\partial x_{i}}\quad\mbox{
for $i=1,2$}.$ (B.2)

###### Proof.

Recall that $u_{p}\in C^{\infty}(\Omega)$ and $Du_{p}\neq 0$ in $\Omega$.
Using equation $-(p-2)\Delta^{N}_{\infty}u_{p}=\Delta u_{p}$ and the identity
in Remark 6.3 one has

${\rm div}(BDw)=0\quad{\rm in}\quad\Omega,$

where

$w=\ln|Du_{p}|,\quad B=(p-2)\frac{Du_{p}\otimes Du_{p}}{|Du_{p}|^{2}}+I_{2}.$

Here $\otimes$ stands for tensor product and $I_{2}$ is identity matrix. Since
$Du_{p}\neq 0$ in $\Omega$, then (B.1) follows from the maximum principle.

To prove (B.2), for each $1\leq j\leq 2$ we differentiate equation (1.20) to
get

$\left(|Du_{p}|^{p-2}\frac{\partial u_{p}}{\partial
x_{i}}\right)_{x_{i}x_{j}}=0\quad{\rm in}\quad\Omega.$

This further leads to

${\rm div}\left(A\frac{\partial u_{p}}{\partial x_{j}}\right)=0\quad{\rm
in}\quad\Omega,$

where

$A=|Du_{p}|^{p-2}\left[(p-2)\frac{Du_{p}\otimes
Du_{p}}{|Du_{p}|^{2}}+I_{2}\right].$

By $Du_{p}\neq 0$ in $\Omega$, thus (B.2) also holds via the maximum
principle. ∎

Denote by $u$ be the $\infty$-harmonic potential in $\Omega$. Recall that
$Du_{p}\to Du$ is locally uniform on $\Omega$ in Theorem 1.1, we have the
following monotonicity property for $u$.

######  Lemma B.2.

For all $B(z,r)\Subset\Omega$, we have

$\displaystyle\max_{\overline{B(z,r)}}|Du|=\max_{\partial
B(z,r)}|Du|,\quad\min_{\overline{B(z,r)}}|Du|=\min_{\partial B(z,r)}|Du|$
(B.3)

and

$\displaystyle\max_{\overline{B(z,r)}}u_{x_{i}}=\max_{\partial
B(z,r)}u_{x_{i}},\quad\min_{\overline{B(z,r)}}u_{x_{i}}=\min_{\partial
B(z,r)}u_{x_{i}}\quad\mbox{ for $i=1,2$}.$ (B.4)

###### Proof.

We only prove $\max_{\overline{B(z,r)}}|Du|=\max_{\partial B(z,r)}|Du|$; the
others can be proved in a similar way. Assume that is not correct. Then one
can find a point $x\in B(z,r)$ such that
$|Du(x)|>\max_{\overline{B(z,r)}}|Du|$. Since $|Du_{p}|\to|Du|$ in
$C^{0}(\overline{B(z,r)})$, for all sufficiently large $p$, we have
$|Du_{p}(x)|>\max_{\overline{B(z,r)}}|Du_{p}|$, which is a contradiction. ∎

Due to Lebesgue in [40], the monotonicity allows us to get the following.

######  Lemma B.3.

For all $B(z,4r)\Subset\Omega$, we have

$\displaystyle\frac{{\rm osc}_{B(z,r)}|Du|^{2}}{r}\leq
C\left(\mathop{}\mkern-3.0mu\mathchoice{\hbox
to0.0pt{$\displaystyle\vbox{\hbox
to18.58415pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\textstyle\vbox{\hbox
to18.58415pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptstyle\vbox{\hbox
to14.39316pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox
to0.0pt{$\scriptscriptstyle\vbox{\hbox
to14.39316pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B_{2r}(z)}|D|Du|^{2}|^{2}\,dx\right)^{\frac{1}{2}}$
(B.5)

and

$\displaystyle\frac{{\rm osc}_{B(z,r)}u_{x_{i}}}{r}$
$\displaystyle\leq\frac{\|\mathcal{D}[u_{x_{i}}]\|(B(z,2r))}{r^{2}}\quad\mbox{
for $i=1,2$}.$ (B.6)

###### Proof.

Let $B(z,4r)\Subset\Omega$ and $0<\epsilon<\frac{1}{4}r$. We set
$g^{\epsilon}=|Du|^{2}\ast\eta^{\epsilon}$ where $\eta^{\epsilon}$ is standard
mollifier. Using polar coordinates in the plane, we have

$\displaystyle{\rm osc}_{\partial
B(z,\rho)}g^{\epsilon}\leq\int^{2\pi}_{0}\frac{\partial
g^{\epsilon}}{\partial\theta}(\rho,\theta)\,d\theta.$

Observing

$\displaystyle|Dg^{\epsilon}|^{2}=\rho^{-2}\left(\frac{\partial
g^{\epsilon}}{\partial\theta}\right)^{2}+\left(\frac{\partial
g^{\epsilon}}{\partial\rho}\right)^{2}\geq\rho^{-2}\left(\frac{\partial
g^{\epsilon}}{\partial\theta}\right)^{2},$

we have

$\displaystyle\int^{2r}_{r}{\rm osc}_{\partial
B(z,\rho)}g^{\epsilon}d\rho\leq\int^{2\pi}_{0}\int^{2r}_{r}\rho^{-1}\frac{\partial
g^{\epsilon}}{\partial\theta}(\rho,\theta)\rho\,d\rho
d\theta\leq\int_{B(z,2r)}|Dg^{\epsilon}|\,dx.$

Since $|Du|\in C^{0}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)\cap
W^{1,2}_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$, by [28, Theorem 6,
Appendix C.4] we have $g^{\epsilon}\to|Du|^{2}$ in $C^{0}(B(z,2r))$ and
$Dg^{\epsilon}\to D|Du|^{2}$ in $L^{2}(B(z,2r))$ as $\epsilon\to 0$. Sending
$\epsilon\to 0$, one gets

$\displaystyle\int^{2r}_{r}{\rm osc}_{\partial
B(z,\rho)}|Du|^{2}\,d\rho\leq\int_{B(z,2r)}|D|Du|^{2}|\,dx.$

Since (B.3) gives

$\mathop{\mathrm{missing}}{\,osc\,}_{B(z,r)}|Du|^{2}\leq\mathop{\mathrm{missing}}{\,osc\,}_{\overline{B(z,\rho)}}|Du|^{2}=\mathop{\mathrm{missing}}{\,osc\,}_{\partial
B(z,\rho)}|Du|^{2},\quad\forall\rho\in[0,2r],$

we have

$\frac{\mathop{\mathrm{missing}}{\,osc\,}_{B(z,r)}|Du|^{2}}{r}\leq\frac{1}{r^{2}}\int^{2r}_{r}{\rm
osc}_{\partial
B(z,\rho)}|Du|^{2}\,d\rho\leq\frac{1}{r}\int_{B(z,2r)}|D|Du|^{2}|\,dx.$

Applying the Hölder inequality, we obtain (B.5) as desired.

Similarly, for $1\leq i\leq 2$, write
$u^{\epsilon}_{x_{i}}=u_{x_{i}}\ast\eta^{\epsilon}$ where $\eta^{\epsilon}$ is
standard mollifier. By an argument similar to above, one has

$\displaystyle\int^{2r}_{r}{\rm osc}_{\partial
B(z,\rho)}u^{\epsilon}_{x_{i}}\,d\rho\leq\int_{B(z,2r)}|Du^{\epsilon}_{x_{i}}|\,dx$

By $u_{x_{i}}\in BV_{{\mathop{\mathrm{missing}}{\,loc\,}}}(\Omega)$ and
$u^{\epsilon}_{x_{i}}=u_{x_{i}}\ast\eta^{\epsilon}$, one has

$\limsup_{\epsilon\to
0}\int_{B(z,2r)}|Du^{\epsilon}_{x_{i}}|\,dx\leq\|\mathcal{D}[u_{x_{i}}]\|(B(z,2r)).$

Using $u_{x_{i}}^{\epsilon}\to u_{x_{i}}$ in $C^{0}(\Omega)$ again,

$\int^{2r}_{r}{\rm osc}_{\partial
B(z,\rho)}u_{x_{i}}\,d\rho\leq\|\mathcal{D}[u_{x_{i}}]\|(B(z,2r)).$

Since (B.4) yields

$\frac{\mathop{\mathrm{missing}}{\,osc\,}_{B(z,r)}u_{x_{i}}}{r}\leq\frac{1}{r^{2}}\int^{2r}_{r}{\rm
osc}_{\partial B(z,\rho)}u_{x_{i}}\,d\rho,$

we obtain (B.6) as desired. ∎

## Acknowledgments

The authors would like to thanks Professor Peter Lindqvist for several
valuable comments and suggestions, in particular, pointing out a mistake in
earlier version.

F. Peng is supported by the National Natural Science Foundation of China
(No.12201612) and the Project funded by China Postdoctoral Science Foundation
(No. BX20220328). Y. Zhang is funded by the Chinese Academy of Science, NSFC
grant No. 12288201, and CAS Project for Young Scientists in Basic Research,
Grant No. YSBR-03. Y. Zhou is supported by NSFC (No.12025102) and by the
Fundamental Research Funds for the Central Universities.

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