Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

Adriano Prade

Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

Adriano Prade

Abstract

For $0<s<1$, we consider the nonlocal equation $(-Δ)^s u = f$ over a Reifenberg flat domain $Ω$ with $f \in C({\overlineΩ})$ and null Dirichlet exterior condition. Given $α\in (0,s)$, we prove that weak solutions are $α$-Hölder continuous up to the boundary when the flatness parameter is small enough. The main ingredients of the proof are an iterative argument and a nonlocal version of the ABP maximum principle.

Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

Abstract

For

, we consider the nonlocal equation

over a Reifenberg flat domain

with

and null Dirichlet exterior condition. Given

, we prove that weak solutions are

-Hölder continuous up to the boundary when the flatness parameter is small enough. The main ingredients of the proof are an iterative argument and a nonlocal version of the ABP maximum principle.

Paper Structure (1 theorem, 29 equations)

This paper contains 1 theorem, 29 equations.

Key Result

Theorem 2

Let $s \in (0,1)$ and $\alpha \in (0,s)$. Let $\Omega \subset \mathbb{R}^n$ be a bounded, open and connected $(\eta, r_0)$-Reifenberg flat domain, $f \in C(\overline{\Omega})$ and $u$ be the weak solution to the system Then, there exists a constant $\eta_0(n,s,\alpha) >0$ such that for all $\eta \leq \eta_0$ we have $u \in C^{\alpha}(\overline{\Omega})$, with the estimate for some constant $C=C(

Theorems & Definitions (4)

Definition 1
Theorem 2
proof
Remark 3