Boundary regularity for a degenerate elliptic equation with mixed boundary conditions

TL;DR

This paper establishes Hölder regularity estimates for solutions to a degenerate elliptic equation with mixed boundary conditions, arising from the Caffarelli-Silvestre extension of the fractional Laplacian, using compactness and blow-up techniques.

Contribution

It provides the first regularity results for solutions with mixed boundary conditions in the degenerate elliptic setting related to fractional Laplacians.

Findings

Hölder regularity of U/d^s near the boundary

Extension of regularity results to mixed boundary conditions

Application of blow-up analysis in degenerate elliptic equations

Abstract

We consider a function U satisfying a degenerate elliptic equation on (0,+\infty)\times R^N with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain \Omega\subset 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;R^N\setminus\Omega), for z\in (0,+\infty)\times 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.