Two overdetermined problems for anisotropic $p$-Laplacian

1.   Introduction

The study of symmetry in overdetermined boundary value problems has become an important field of research in the theory of PDEs. The pioneering symmetry result obtained by Serrin ^[20] are now classical but still influential. The main technique to tackle such problems are the celebrated method of moving planes developed by Alexandrov ^[2,3] and Serrin ^[20] as well as Weinberger's approach ^[25] which is based on maximum principle for so-called $P$-function and Rellich-Pohozaev's integral identity.

There are plenty of considerations for different kinds of overdetermined boundary value problems. For our purpose, we recall a result of Reichel ^[17], who considered an overdetermined problem for capacity in an exterior domain. The capacity of a smooth bounded domain ${\Omega}\subset {\mathbb{R}}^n (n\ge 2)$ is defined as

The minimizer for ${\rm Cap}({\Omega})$ is characterized by the capacitary potential $u$ satisfying

Reichel ^[17] considered the overdetermined problem, (1.1) with an extra boundary condition

and proved that (1.1) and (1.2) admits a solution if and only if ${\Omega}$ is a ball. Reichel's proof is again based on the method of moving planes and he also extended in ^[18] such result to more general quasilinear equations including $p$-Laplacian equations in exterior domains. Garofalo-Sartori ^[10] and Poggesi ^[14] reproved Reichel's result for $p$-capacity by using Weinberger type approach, which was first used by Payne-Philippin ^[16] for the exterior problem.

The anisotropic PDE problems involving the anisotropic Laplacian attract lots of attention in recent decades. Regarding the overdetermined problem, Cianchi-Salani ^[6] and Wang-Xia ^[22] independently extends Serrin's classical result in the anisotropic setting. Due to the anisotropy, the method of moving planes does not work but Weinberger type approach works in general. The correponding overdetermined problem for anisotropic $p$-capacity in an exterior domain considered by Reichel ^[17,18] has been extended by Bianchini-Ciraolo ^[4] and Bianchini-Ciraolo-Salani ^[5]. They proved the symmetric result when the domain is assumed to be convex, by using a totally integral method. In this paper, we remove the convexity assumption in Bianchini-Ciraolo-Salani's result by using Weinberger type approach.

In order to state our result, we introduce the anisotropic $p$-capacity. Let $F\in C^\infty({\mathbb{R}}^n\setminus \{0\})$ be a Minkowski norm in ${\mathbb{R}}^n$, see Section 2.1 for the definition. For $p\in (1, n)$, the anisotropic $p$-capacity of $\Omega$ is defined as

The associated anisotropic $p$-capacitary potential is namely the unique weak solution $u$ to the following problem

where $\Delta_{F, p}$ is the anisotropic (Finsler-)$p$-Laplacian,

A function $u\in W^{1, p}_{loc}(\mathbb R^n\setminus\bar{\Omega})$ is called a weak solution of $\Delta_{F, p}u = 0$ in $\mathbb R^n\setminus\bar{\Omega}$ if

for any $\psi\in C^{\infty}_{c}(\mathbb R^n\setminus\bar{\Omega})$. It is not hard to see that

where $\nu = -\frac{{\nabla} u}{|{\nabla} u|}$ is a unit normal of ${\partial} {\Omega}$ pointing towards $\mathbb R^n\setminus\bar{\Omega}$.

We will study the problem (1.4) with the overdetermined condition

for some constant $c > 0$. The first main result in this paper is the following

Theorem 1.1. Let $1 < p < n$ and $\Omega\subset\mathbb R^n$ be a bounded domain with boundary of class $C^{2, \alpha}$. Then (1.4) and (1.5) admits a weak solution if and only if $\Omega$ is a Wulff ball.

In the theorem, a Wulff ball means a translation and rescaling of $\mathcal{W}_F = \{x\in{\mathbb{R}}^n: F^o(x) < 1\}$, where $F^o$ is the dual norm of $F$ given by (2.1).

Next, we prove a similar result corresponding to Theorem 1.1 in the special case $p = n$. Let $u\in W^{1, n}(\mathbb R^n\setminus\bar\Omega)$ be a weak solution to

where $\thicksim$ means that

for some positive constant $c_1, c_2$. For this case, we prove the following result with analogous tools.

Theorem 1.2. Let $\Omega\subset\mathbb R^n$ be a bounded domain with boundary of class $C^{2, \alpha}$. Then (1.6) and (1.5) admit a weak solution if and only if $\Omega$ is a Wulff ball.

We adapt the arguments in Garofalo-Sartori ^[10] and Poggesi ^[14] to prove Theorems 1.1 and 1.2. The main ingredients are a strong maximum principle on a well-behaved $P$-function and a Rellich-Pohozaev-type identity.

In the second part of this paper, we consider a similar overdetermined problem for the anisotropic $p$-Laplacian in a bounded punctured domain. More precisely, we are concerned with the following equation in $\Omega\setminus\{0\}$, where $0$ is contained in $\Omega$: for $p\in (1, n],$

under Serrin's overdetermined condition

We say a function $u\in W^{1, p}_{loc}({\Omega}\setminus\{0\})$ is a weak solution of $\Delta_{F, p}u = 0$ in ${\Omega}\setminus\{0\}$ if

for any $\psi\in C^{\infty}_{c}({\Omega}\setminus\{0\})$. $\lim_{|x|\rightarrow0} u(x) = +\infty$ means $u$ has a non-removable singularity at $0$. A classical result of Serrin ^[19] says that for $p\in(1, n]$, when $u$ has a non-removable singularity at $0$, then $\frac{u}{\Gamma_{F, p}}$ is bounded near $0$, where $\Gamma_{F, p}$ is the fundamental solution to the anisotropic $p$-Laplacian given in (3.1).

Theorem 1.3. Let $1 < p \leq n$ and $\Omega\subset\mathbb R^n$ be a bounded domain with boundary of class $C^{2, \alpha}$. Then (1.8) and (1.9) admits a weak solution if and only if $\Omega$ is a Wulff ball centered at $0$.

When $F$ is the Euclidean norm, such symmetric result has been proved by Alessandrini-Rosset ^[1], and Enciso-Peralta-Salas ^[9] via the method of moving planes and Weinberger type approach, respectively. We shall adapt Enciso-Peralta-Salas's method ^[9] which is based on the $P$-function to prove Theorem 1.3.

Throughout this paper, we assume that $F\in C^{\infty}({\mathbb{R}}^{n}\setminus\{0\})$ is a Minkowski norm on ${\mathbb{R}}^n$, and ${\Omega}$ is a bounded domain with boundary of class $C^{2, \alpha}$. We will always use Einstein summation convention.

2.   Preliminaries

2.1.   Minkowski norm, Wulff shape, anisotropic area

Let $F\in C^{\infty}({\mathbb{R}}^{n}\setminus\{0\})$ be a Minkowski norm on ${\mathbb{R}}^n$, in the sense that

$\rm(i)$ $F$ is a norm in $\mathbb{R}^{n}$, i.e., $F$ is a convex, $1$-homogeneous function satisfying $F(x) > 0$ when $x\neq 0$ and $F(0) = 0$;

$\rm(ii)$ $F$ satisfies a uniformly elliptic condition: ${\nabla}^2 (\frac12 F^2)$ is positive definite in $\mathbb{R}^{n}\setminus \{0\}$.

The dual norm $F^o\colon {\mathbb R^n} \rightarrow [0, +\infty[$ of $F$ is defined as

$F^o$ is also a Minkowski norm on ${\mathbb{R}}^{n}$. Furthermore,

We remark that, throughout this paper, we use conventionally $\xi$ as the variable for $F$ and $x$ as the variable for $F^o$.

Denote

For the simplicity of notations, we will denote by $\mathcal W_F = \mathcal W$. We call $\mathcal W$ the unit Wulff ball centered at the origin, and ${\partial} \mathcal W$ the Wulff shape.

More generally, we denote

and call it the Wulff ball of radius $r$ centered at $x_0$. We simply denote $\mathcal W_r = \mathcal W_r(0)$.

The following properties of $F$ and $F^o$ hold true and will be frequently used in this paper (see e.g., ^[7,24]).

Proposition 2.1. Let $F:\mathbb R^n\rightarrow [0, \infty)$ be a Minkowski norm. Then for any $x, \xi \in {\mathbb R^n}\setminus \{0\}$, the following hold:

1). $\langle F_{\xi} (\xi), \xi \rangle = F(\xi), \quad \langle F^{o}_x (x), x \rangle = F^{o}(x).$

2). $\sum_jF_{\xi_i\xi_j}(\xi)\xi_j = 0$ for any $i = 1, \ldots, n$.

3). $F(F_{x}^o(x)) = F^o(F_{\xi}(\xi)) = 1.$

4). $F^o(x) F_{\xi}(F_x^o(x)) = x, \quad F(\xi) F_{x}^o\left(F_{\xi}(\xi) \right) = \xi.$

Let $\Omega\subset\mathbb R^n$ be a bounded open set with smooth boundary ${\partial}{\Omega}$ and $\nu$ be its unit outward normal of ${\partial}{\Omega}$. The anisotropic area $|{\partial} \Omega|_F$ of ${\Omega}$ is defined by

The well-known Wulff theorem (see e.g., Theorem 20.8 in ^[15]) says that Wulff balls are the only minimizers for the anisotropic isoperimetric problem. Equivalently, the Wulff inequality holds true:

Equality in (2.3) holds if and only if $\Omega$ is a Wulff ball.

Note that when ${\Omega} = \mathcal{W}$, the unit Wulff ball, one can check by the divergence theorem that

For notation simplicity, we denote

2.2.   Anisotropic p-Laplacian

Let $u$ be twice continuous differentiable at $x\in {\mathbb{R}}^n$. We denote by $F_i, F_{ij}, \ldots$ the partial derivatives of $F$ and by $u_i, u_{ij}, \ldots$ the partial derivatives of $u$,

For $x$ such that ${\nabla} u(x)\neq 0$, denote

The anisotropic Laplacian and p-Laplacian of $u$ is given by

2.3.   Anisotropic curvature for level sets

We recall the concept of anisotropic curvature for a hypersurface in $\mathbb R^n$. See e.g., ^[22,26].

Let $M$ be a smooth embedded hypersurface in $\mathbb R^n$ and $\nu$ be one unit normal of $M$. The corresponding anisotropic normal of $M$ is defined by

The anisotropic principal curvatures ${\kappa}_F = (\kappa_{1}^{F}, \ldots, \kappa_{n-1}^{F})\in{\mathbb{R}}^{n-1}$ are defined as the eigenvalues of the map

The mean curvature (with respect to $\nu$) is defined to be

In this paper we are interested in the case when $M$ is given by a regular level set of a smooth function $u$, that is $M = \{u = t\}$ for some regular value $t$. For our purpose, we choose the unit normal $\nu = -\frac{\nabla u}{|\nabla u|}$ and

In this case, we have that

Here $\rm div$ is the Euclidean divergence. See e.g., ^[8].

Next we give the formula for the anisotropic mean curvature of regular level sets.

Proposition 2.2 (^[27]). Let $u$ satisfy $\Delta_{F, p}u = 0$. Then the anisotropic mean curvature of regular level set of $u$ is given by

Finally, we will give the anisotropic Heintze-Karcher inequality for later use.

Proposition 2.3 (^[12,28]). Let $\Omega\subset\mathbb R^n$ be an open bounded domain with $C^{2}$ boundary $\partial{\Omega}$ satisfying $H_F > 0$. Then,

and equality holds if and only if ${\Omega}$ is a Wulff ball.

3.   Overdetermined problem in an exterior domain

3.1.   Regularity and asymptotic behavior

Let $u$ be a weak solution to (1.4) (case $1 < p < n$) or (1.5) (case $p = n$). The following regularity result is nowadays standard by the regularity theory for degenerate elliptic PDEs ^[21] and Schauder theory for uniformly elliptic PDEs ^[11].

Proposition 3.1 (Regularity). Let $1 < p\le n$ and $u$ be a weak solution to (1.4). Then $u\in C^{1, \beta}(\mathbb R^n\setminus{\Omega})$ for some $\beta < 1$. Moreover, $u\in C^{\infty}\big((\mathbb R^n\setminus\bar{\Omega})\setminus{\rm Crit}(u)\big)\cap C^{2, \alpha}\big((\mathbb R^n\setminus {\Omega})\setminus{\rm Crit}(u)\big)$, where ${\rm Crit}(u) = \big\{x\in\mathbb R^n\setminus{\Omega}\, |\, {\nabla} u(x) = 0\big\}$. Moreover, $|\nabla u|\neq 0$ in some neighborhood $\mathcal N({\partial}{\Omega})$ of ${\partial}{\Omega}$ and $u$ is $C^2(\mathcal N({\partial}{\Omega})\cap (\mathbb R^n\setminus{\Omega}))$.

For $1 < p\le n$, let

One can check that

where $\delta_0$ is the Dirac Delta function about the origin. We call $\Gamma_{F, p}$ the fundamental solution to $\Delta_{F, p}u = 0$ in ${\mathbb{R}}^n$. See ^[23].

Proposition 3.2 (Asymptotic behavior, $1 < p < n$ ^[27]).

Let $1 < p < n$ and $u$ is a weak solution to $\Delta_{F, p}u = 0$ in ${\mathbb{R}}^n\setminus \bar {\Omega}$. Then

1). $\lim\limits_{|x|\rightarrow +\infty}\frac{u(x)}{\Gamma_{F, p}(x)} = {{\rm Cap}_{F, p}}(\Omega)^{\frac1{p-1}},$

2). $\nabla u(x) = \, {{\rm Cap}_{F, p}}(\Omega)^{\frac1{p-1}}\nabla \Gamma_{F, p}(x)+o(|x|^{-\frac{n-1}{p-1}}).$ where ${{\rm Cap}_{F, p}}(\Omega)$ is the anisotropic $p$-capacity of $\Omega$ given in (1.3).

Proposition 3.3 (Asymptotic behavior, $p = n$).

Let $p = n$ and $u$ be a weak solution to $(1.6)$ and $(1.5)$. Then

1). $\lim\limits_{|x|\rightarrow +\infty}\frac{u(x)}{\Gamma_{F, n}(x)} = c|\partial\Omega|^{\frac1{n-1}}_F,$

2). $\nabla u(x) = \, c|\partial\Omega|^{\frac1{n-1}}_F\nabla \Gamma_{F, n}(x)+o(|x|^{-1}).$

Proof. If $u$ is solution of (1.6), it is a standard argument by using comparison theorem to show that there exists two positive constants $c_1, c_2$ such that

Following the argument of ^[13], Theorem 1.1 and Remark 1.5, (see ^[23], Theorem 4.1 and Remark 4.1 for anisotropic case), we conclude that there exists $\gamma\in {\mathbb{R}}$ such that

By integration by parts for (1.5), we have

where $\nu = -\frac{\nabla u}{|{\nabla} u|}$ and $\nu_{{\partial}\mathcal{W}_{R}}$ is outward normal vector of $\mathcal{W}_{R}$. From (3.2), we have

On $\partial\mathcal{W}_{R}$,

Hence

Combining the fact that

we deduce that

It follows from (1.5) and (3.4) that

The assertion follows.

Proposition 3.4 (^[27]). Let $1 < p < n$ and $u$ be a weak solution to (1.4). Then

3.2.   Rellich-Pohozaev-type identity

Firstly, we prove the following Rellich-Pohozaev-type identity.

Proposition 3.5. Let $1 < p < n$ and $u$ be a weak solution to (1.4). Then

where $\nu = -\frac{{\nabla} u}{|{\nabla} u|}$ is a unit normal of ${\partial} {\Omega}$ pointing towards $\mathbb R^n\setminus\bar{\Omega}$.

Proof. By directing computations, we get, for $R$ large,

Then, by taking the limit for $R\rightarrow +\infty$ and noting that the integrals on $\partial\mathcal{W}_R$ converge to zero due to the asymptotic behavior of $u$ at infinity given by Proposition 3.2. Thus, we obtain the assertion.

Proposition 3.6. Let $p = n$ and $u$ be a weak solution to (1.6). Then we have

where $X$ is the vector field given by

Proof. The proof is the same as that of Proposition 3.5 by letting $p = n$. We omit it here.

3.3.   Case $1 < p < n$

First we can compute the value $c$ of $F(\nabla u)$ on $\partial{\Omega}$ with the overdetermined condition (1.5).

Proposition 3.7. Let $1 < p < n$ and $u$ be a weak solution to (1.4) and (1.5). The constant $c$ appearing in (1.5) equals

Moreover, the following explicit expression of the anisotropic $p$-capacity of $\Omega$ holds:

Proof. By using (3.5) and (3.6), we obtain that

which implies (3.9) and (3.10).

Next, we introduce the $P$-function

We show that $P$ satisfies a strong maximum principle.

Proposition 3.8. Let $1 < p < n$ and $u$ be a weak solution to (1.4). Then, at $\{{\nabla} u\not = 0\}$,

where

and $L_iP_i$ is lower order term of $P_i$.

Moreover, $P$ cannot attain a local maximum at an interior point of $\mathbb R^n\setminus\bar\Omega$, unless $P$ is a constant.

Proof. Set ${\rm Crit}(u) = \{x\in\mathbb R^n\setminus\bar{\Omega}\, |\, {\nabla} u = 0\}$. The following calculations are all taken in $\big(\mathbb R^n\setminus\bar{\Omega}\big)\setminus{\rm Crit}(u)$.

First we calculate the first and second derivatives of the $P$-function.

The first and the second derivatives of $P$ are

It follows from (3.12) and Proposition 2.1 (1) that

The first equation of (1.4) implies that

(3.15) and (3.16) give us

By taking derivative of (3.16), we obtain

From Proposition 2.1 (2), we have

Taking derivative of (3.19) w.r.t. $x_i$ and summing, we obtain

From Proposition 2.1, (3.12)–(3.20) and (2.5), we have the following computation

in particular,

and, by using (3.20)

From Proposition 2.3 in ^[27], we have

Substituting (3.22)–(3.24) into (3.21), we obtain

This combing with (3.15) yields to

Let $-L_iP_i$ denote the term with $P_i$ in (3.26). We have

If $P$ attains a local maximum at some interior point $x_0\in {\mathbb{R}}^n\setminus \bar{{\Omega}}$, then $x_0\in ({\mathbb{R}}^n\setminus \bar{{\Omega}})\setminus {\rm Crit}(u)$, or $P\equiv 0$ which is impossible. By using the strong maximum principle for (3.27) on a neighborhood $\mathcal N$ of $x_0$ in which $\nabla u\neq 0$, one sees $P$ must be a constant on $\mathcal N$. It follows that the set where $P$ is a constant is both open and closed. Thus $P$ is a constant in ${\mathbb{R}}^n\setminus {\Omega}$.

Proof of Theorem 1.1. By using Proposition 3.2 we can check that

This combining with (3.10) yields to

On the other hand, by using the boundary conditions (1.4), (1.5) and (3.9) we get

According to the Wulff inequality (2.3), by using (3.28) and (3.29), we check that

From Proposition 3.8 we see that either $P$ is a constant or $P$ attains its maximum on ${\partial} {\Omega}$. In both cases, we have

We remark that here $\nu_F = F_\xi(\nu)$ with $\nu = -\frac{\nabla u}{|\nabla u|}$ which points towards ${\mathbb{R}}^n\setminus \bar {\Omega}$. By direct computation we see

In the last equality we have used (2.8). By using fact that $u = 1$ on $\partial\Omega$, (1.5) and (3.9), it follows from (3.30) and (3.31) that

It follows from (3.32) that $H_F$ is positive and

Combining (3.33) with Proposition 2.3, we conclude ${\Omega}$ is a Wulff ball. This completes the proof of Theorem 1.1.

3.4.   Case $p=n$

Proof of Theorem 1.2. We look at the identity (3.7). For the left side of (3.7), by using (1.5), we deduce that

For the right side of (3.7), by the asymptotic behavior in Proposition 3.3, we can easy compute that

It follows that

That means the equality in the Wulff inequality (2.3) holds. Thus ${\Omega}$ is a Wulff ball. This completes the proof of Theorem 1.2.

4.   Overdertermined problem in a punctured domain

In this section, we consider the overdetermined problem (1.8) and (1.9) in a bounded punctured domain.

Since $0$ is non-removable singular point, by using a result of Serrin ^[19], we can see that $u/\Gamma_{F, p}$ is bounded in some neighborhood of $0$. Moreover we can see that if $\Delta_{F, p}u = 0$ in $\Omega\setminus\{0\}$, then $u$ satisfies (see ^[13] or Theorem 4.1 in ^[23])

where $\delta_0$ is the Dirac measure in the origin and $K$ is some non-zero constant.

Proposition 4.1 (Regularity and asymptotic behavior). Let $1 < p\leq n$ and $u$ be a weak solution to (1.8) and (1.9). Then

1). $u\in C^{1, \beta}(\bar{\Omega}\setminus\{0\})$ for some $\beta < 1$. Moreover, $u\in C^{\infty}\big(\Omega\setminus (\{0\}\cup{\rm Crit}(u)\big))\cap C^{2, \alpha}\big(\bar{\Omega}\setminus (\{0\}\cup{\rm Crit}(u)\big))$, where ${\rm Crit}(u) = \big\{x\in\bar \Omega\, |\, {\nabla} u(x) = 0\big\}$. Moreover, $|\nabla u|\neq 0$ in some neighborhood $\mathcal N({\partial}{\Omega})$ of ${\partial}{\Omega}$ and $u$ is $C^2(\mathcal N({\partial}{\Omega})\cap \bar \Omega)$.

2). The constant $K$ that appears in (4.1) satisfies $K = c^{p-1}|\partial\Omega|_F$.

3). Asymptotic behavior of the solution $u$ of (1.8) near the origin is given by

(a) $\lim\limits_{|x|\rightarrow +\infty}\frac{u(x)}{\Gamma_{F, p}(x)} = c|\partial\Omega|^{\frac1{n-1}}_F,$

(b) $\nabla u(x) = \, c|\partial\Omega|^{\frac1{p-1}}_F\nabla \Gamma_{F, p}(x)+o(|x|^{-\frac{n-1}{p-1}}).$

Proof. We prove (2). Let $U$ and $V$ be tubular neighborhoods of $\partial\Omega$ such that $\nabla u\neq 0$ in $U\cap\Omega$ and $V\subset\subset U$. Hence $u\in C^2(U\cap\Omega)$, and

We choose a function $\phi_{+}$ with $\phi_{+} = 1$ on ${\Omega}\setminus U$ and ${\rm supp}\, \phi_{+}\subset\Omega\setminus\bar V$, and define $\phi_{-}: = 1-\phi_{+}$. From (4.2), (1.9) and (4.1), we have

For (3), the proof is similar with that of that of Propositions 3.2 and 3.3. We omit it here.

Proof of Theorem 1.3: $1 < p < n$. Firstly, note that the actual value of the function $F({\nabla} u)$ on the boundary is irrelevant, because for any arbitrary constant $c_0\not = c$, there exists another solution $\tilde u = \frac{c_0}{c}(u-1)+1$ satisfying $\tilde u = 1$ and $\Delta_{F, p}\tilde u = 0$ and $F(\nabla \tilde{u}) = c_0$. Hence, without loss of generality one can set

By (4.3) and Proposition 4.1 (3), the maximum principle for the anisotropic $p$-Laplacian yields that $u > 0$ in $\Omega\setminus\{0\}$.

Next, we define the $P$-function exactly as in the exterior case, that is,

Then we get as in Proposition 3.8 that

Hence, $P$ cannot attain a local maximum at an interior point of ${\Omega}\setminus\{0\}$, unless $P$ is a constant. On the other hand, by Proposition 4.1 (3), we obtain

Moreover, from (4.3) and the boundary condition, one can check that

It follows that

Now, we shall show that $P$ is actually a constant. It follows from (4.1) that

for $\tilde{\phi}\in C^{1}_{c}(\Omega)$. Let $\tilde\phi = \phi u^{\alpha}$ for $\phi\in C^{\infty}_{c}(\Omega)$ and some $\alpha < 0$ to be fixed. From the asymptotic behavior of $u$ near $0$ and $u\in C^{1}(\bar {\Omega}\setminus\{0\})$, we see $\tilde\phi\in C^1_{c}(\Omega)$. Using $\tilde{\phi}$ in (4.4) and let $\alpha = -n\frac{p-1}{n-p}$, we see

Also, near ${\partial} {\Omega}$, we have $\Delta_{F, p}u = 0$ which yields to

in the classical sense. By using the same $\phi_+$ and $\phi_{-}$ stated in the proof of Proposition 4.1, we have

Furthermore, by using (1.8), (1.9) and (4.3), we obtain

where the last inequality follows directly from the Wulff inequality (2.3). One sees that the equality holds in (4.7) which implies that $\Omega$ is Wulff ball by the equality characterization in the Wulff inequality.

Proof of Theorem 1.3: $p = n$. In this case, we use similar method as the proof of Theorem 1.2. We also have a similar integral equality as (3.7),

where $\nu = -\frac{{\nabla} u}{|{\nabla} u|}$ is the unit outward normal of ${\Omega}$ and $X$ is given by (3.8).

For the left hand side of (4.8), by using (1.9), we have

For the right side of (4.8), by Proposition 4.1 (3), we can compute that

It follows that

which is the equality in the Wulff inequality (2.3). It follows that ${\Omega}$ is a Wulff ball. This completes the proof of Theorem 1.3.

Acknowledgments

CX was supported by NSFC (Grant No. 11871406) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180009).

Conflict of interest

The authors declare no conflict of interest.