On the boundary H\"{o}lder regularity for the infinity Laplace equation

TL;DR

This paper establishes boundary Hölder regularity for the infinity Laplace equation under a broad geometric condition, extending previous results to various domain types using maximum principles and scaling invariance.

Contribution

It introduces a general geometric condition ensuring boundary Hölder regularity for the infinity Laplace equation, encompassing several classical domain types.

Findings

Boundary Hölder regularity proven under general geometric conditions

Applicable to exterior cone, Reifenberg flat, and corkscrew domains

Utilizes maximum principle and scaling invariance techniques

Abstract

In this note, we prove the boundary H\"{o}lder regularity for the infinity Laplace equation under a proper geometric condition. This geometric condition is quite general, and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The key idea, following [3], is that the strong maximum principle and the scaling invariance imply the boundary H\"{o}lder regularity.

Equations

$$\{\begin{array}{rl}{\Delta }_{\infty }u=u_i{u}_j{u}_{ij}=0& \text{ in }\Omega ;\\ u=g& \text{ on }\partial \Omega ,\end{array}$$

$${\tau }_1{r}_{k-1}\le r_k\le {\tau }_2{r}_{k-1},\forall k\ge 1,$$

$$\partial B(x_0,r_k)\cap B(y_k,\nu r_k)\subset {\Omega }^c.$$

$$|g(x)-g(x_0)|\le K|x-x_0{|}^{\alpha },\forall x\in \bar{\Omega }.$$

$$\{\begin{array}{rl}{\Delta }_{\infty }u=0& \text{ in }\Omega \cap B_1;\\ u=g& \text{ on }\partial \Omega \cap B_1,\end{array}$$

$$|u(x)-u(0)|\le 8|x{|}^{\beta }\left(\Vert u{\Vert }_{L^{\infty }(\Omega \cap B_1)}+[g{]}_{C^{\alpha }(0)}\right),\forall x\in \Omega \cap B_1,$$

$$\{\begin{array}{rl}{\Delta }_{\infty }u=0& \text{ in }\Omega ;\\ u=g& \text{ on }\partial \Omega .\end{array}$$

$${\Delta }_{\infty }u=0\text{ in }{B}_1.$$

$$\underset{B_{1/2}}{\sup }u\le 3\underset{B_{1/2}}{\inf }u.$$

$$\underset{{\Omega }^{\prime }}{\sup }u\le (1-\mu )\underset{\Omega }{\sup }u,$$

$$g_{\nu }(x)\equiv \{\begin{array}{rlrl}& 0& & \text{mbox}on\partial B_1\cap B(e_1,\nu /2);\\ & 1& & \text{mbox}on\partial B_1\backslash B(e_1,\nu ),\end{array}$$

$$g_k(x)\equiv \{\begin{array}{rlrl}& 0& & \text{mbox}on\partial B_{r_k}\cap B(y_k,\nu r_k/2);\\ & 1& & \text{mbox}on\partial B_{r_k}\backslash B(y_k,\nu r_k)\end{array}$$

$$g_k(r_k\cdot T_{k}x)\equiv g_{\nu }(x)\text{mbox}on\partial B_1$$

$$\{\begin{array}{rlrl}{\Delta }_{\infty }v& =0& & \text{ in }{B}_{r_k};\\ v& =a{g}_k+b& & \text{ on }\partial B_{r_k},\end{array}$$

$$\underset{B_{r_{k+1}}}{\sup }v\le (1-\mu )a+b,$$

$$w(y)=\frac{v(x)-b}{a}.$$

$$\{\begin{array}{rlrl}{\Delta }_{\infty }w& =0& & \text{ in }{B}_1;\\ w& =g_{\nu }& & \text{ on }\partial B_1.\end{array}$$

$$\underset{B_{{\tau }_2}}{\sup }w\le (1-\mu ),$$

$$\underset{B_{r_{k+1}}}{\sup }v\le \underset{B_{{\tau }_2{r}_k}}{\sup }v\le (1-\mu )a+b.$$

$${\tau }_1^{\beta }\ge \frac{1}{2}$$

$$\Vert u{\Vert }_{L^{\infty }({\Omega }_{r_k})}\le 4M{r}_k^{\beta }.$$

$$|u(x)|\le 4M{r}_k^{\beta }\le \frac{4M{r}_{k+1}^{\beta }}{{\tau }_1^{\beta }}\le 8M|x{|}^{\beta }.$$

$$\{\begin{array}{rl}{\Delta }_{\infty }v=0& \text{ in }{B}_{r_k};\\ v=\tilde{g}& \text{ on }\partial B_{r_k},\end{array}$$

$$-v\le u\le v\text{mbox}on\partial {\Omega }_{r_k}.$$

$$-v\le u\le v\text{ in }{\Omega }_{r_k}.$$

$$\underset{B_{r_{k+1}}}{\sup }v$$

$$=(1-\mu )4M{r}_k^{\beta }+\mu M{r}_k^{\beta }$$

$$\le 4M{r}_{k+1}^{\beta }\left(\frac{1-\mu }{{\tau }_1^{\beta }}+\frac{\mu }{4{\tau }_1^{\beta }}\right),$$

$$\frac{1-\mu }{{\tau }_1^{\beta }}<1-\frac{\mu }{2}.$$

$$\underset{B_{r_{k+1}}}{\sup }v\le 4M{r}_{k+1}^{\beta }.$$

$$\Vert u{\Vert }_{L^{\infty }({\Omega }_{r_{k+1}})}\le 4M{r}_{k+1}^{\beta }.$$

Full text

In this note, we prove the boundary Hölder regularity for the infinity Laplace equation:

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain, $u_{i}=\partial u/\partial x_{i}$, $u_{ij}=\partial^{2}u/\partial x_{i}\partial x_{j}$ and the Einstein summation convention is used.

The following boundary regularity are well known. If $g\in C^{0,1}(\partial\Omega)$ (i.e., $g$ is Lipschitz continuous on $\partial\Omega$), $u\in C^{0,1}(\bar{\Omega})$ (i.e., $u$ is Lipschitz continuous on $\bar{\Omega}$). In addition, if $g\in C^{0}(\partial\Omega)$, $u\in C^{0}(\bar{\Omega})$. It should be noted that both results hold without any geometric condition on $\partial\Omega$.

It is natural to study the boundary Hölder regularity, i.e., whether $g\in C^{\alpha}(\partial\Omega)$ implies $u\in C^{\alpha}(\bar{\Omega})$. This is the aim of this note. To the authors’ knowledge, our result is the first one contributing to the boundary Hölder regularity.

In this note, we obtain the boundary Hölder regularity for the solutions of0.1 under a proper geometric condition on $\partial\Omega$. The idea and the method originate from [3]. As pointed out in [3], this geometric condition is quite general and the exterior cone condition, the Reifenberg flat domains, and the corkscrew domains (including the non-tangentially accessible domains) are special cases. The main idea is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Hölder regularity.

The following is the geometric condition under which we prove the boundary Hölder regularity.

[TABLE]

and for any $k\geq 0$, there exists $y_{k}\in\partial B(x_{0},r_{k})$ such that

[TABLE]

We say that $g$ is $C^{\alpha}$ at $x_{0}\in\bar{\Omega}$ or $g\in C^{\alpha}(x_{0})$ if there exists a constant $K$ such that

[TABLE]

Then denote $[g]_{C^{\alpha}(x_{0})}=\inf K$.

Our main result is the following.

[TABLE]

where $g\in C^{\alpha}(0).$

Then $u$ is $C^{\beta}$ at [math] and

[TABLE]

where $0<\beta\leq\alpha$ depends only on $n,\tau_{1},\tau_{2}$ and $\nu$.

The proof of Theorem 0.2 depends on the solvability and the strong maximum principle for the infinity Laplace equation, which are already known. The following lemma shows the solvability (see Theorem 5.21 in [2]).

[TABLE]

The strong maximum principle can be derived easily from the following Harnack inequality (see Theorem 2.21 in [2] or Proposition 6.3 in [1]).

[TABLE]

Then

[TABLE]

[TABLE]

where $\Omega^{\prime}\subset\subset\Omega$ and $\mu$ depends only on $\Omega,\Omega^{\prime}$ and $g$.

Let $0<\nu<1$ be as in the uniform condition. Then we choose and fix a function $g_{\nu}\in C^{\infty}(\partial B_{1})$ with $0\leq g_{\nu}\leq 1$ and

[TABLE]

where $e_{1}=(1,0,0,...,0)$. Next, introduce functions $g_{k}$ ($k\geq 0$) with

[TABLE]

and

[TABLE]

for some orthogonal matrix $T_{k}$. Here $\{r_{k}\}$ and $\{y_{k}\}$ are as in the uniform condition. From the strong maximum principle, we have the following simple result.

[TABLE]

where $a,b>0$.

Then

[TABLE]

where $0<\mu<1$ depends only on $n,\tau_{2}$ and $\nu$.

[TABLE]

Then $w$ satisfies

[TABLE]

Then by the strong maximum principle (see Lemma 0.4)

[TABLE]

where $0<\mu<1$ depends only on $n,\tau_{2}$ and $g_{\nu}$. Note that $g_{\nu}$ depends only on $\nu$. Hence, $\mu$ depends only on $n,\tau_{2}$ and $\nu$. By rescaling,

[TABLE]

∎

Now we give the

Proof of Theorem 0.2. We assume that $g(0)=0$. Otherwise, we may consider $v=u-g(0)$. Let $M=\|u\|_{{L^{\infty}}(\Omega\cap B_{1})}+[g]_{C^{\alpha}(0)}$ and $\Omega_{r}=\Omega\cap B_{r}.$ To prove0.4, we only need to prove the following:

There exists a constant $0<\beta\leq\alpha$ depending only on $n,\tau_{1},\tau_{2}$ and $\nu$ such that

[TABLE]

and for all $k\geq 0,$

[TABLE]

Indeed, suppose that0.8 and0.9 hold. Then for any $x\in\Omega\cap B_{1}$, there exists $k$ such that $r_{k+1}\leq|x|\leq r_{k}$. Hence,

[TABLE]

We prove0.9 by induction. For $k=0,$ it holds clearly. Suppose that it holds for $k,$ then we need to prove that it holds for $k+1.$

By Lemma 0.3, there exists a unique viscosity solution $v$ of

[TABLE]

where $\tilde{g}=(4Mr_{k}^{\beta}-Mr_{k}^{\alpha})g_{k}+Mr_{k}^{\alpha}.$ Then it is easy to check that

[TABLE]

Then by the comparison principle (see [2, Theorem 5.22]),

[TABLE]

From Lemma 0.6, we have

[TABLE]

where $0<\mu<1$ depends only on $n,\tau_{2}$ and $\nu.$

Take $\beta$ small enough such that0.8 holds and

[TABLE]

Hence,

[TABLE]

Then, combining with0.10, we have

[TABLE]

By induction, the proof is completed. ∎