Published Paper

Inserted: 26 jul 2019
Last Updated: 26 jul 2019

Journal: Communications on Pure and Applied Analysis
Volume: 18
Number: 1
Pages: 22
Year: 2019
Doi: doi:10.3934/cpaa.2019007
Links: African Institute for Mathematical Sciences

Abstract:

We consider a function $U$ satisfying a degenerate elliptic equation on $(0,+\infty) \times \mathbb{R}^N$ with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain $\Omega$ of $\mathbb{R}^N$ of class $C^{1,1}$, whereas the Dirichlet data is on the exterior of $\Omega$. We prove Holder regularity estimates of $U/d^s$, where $d$ is a distance function defined as $d(z) := dist(z,\mathbb{R}^N\setminus\Omega)$, for $z\in(0,+\infty)\times \mathbb{R}^N$. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments.

Keywords: fractional Laplacian, boundary regularity, blow-up analysis, Caffarelli-Silvestre extension