Boundary blow-up and degenerate equations

TL;DR

This paper proves that solutions to a specific degenerate elliptic PDE with boundary blow-up are regular up to the boundary, using new Schauder estimates for Fuchsian type equations.

Contribution

It introduces novel Schauder estimates for degenerate elliptic equations of Fuchsian type, establishing boundary regularity for solutions with blow-up behavior.

Findings

Solutions are of class C^{2+α} up to the boundary.

New Schauder estimates for degenerate elliptic equations are developed.

Boundary blow-up solutions exhibit regularity similar to non-degenerate cases.

Abstract

Let $\Omega\subset\mathbb R^2$ be a bounded domain of class $C^{2+\alpha}$, $0<\alpha<1$. We show that if $u$ is the solution of $\Delta u = 4\exp(2u)$ which tends to $+\infty$ as $(x,y)\to\partial\Omega$, then the hyperbolic radius $v=\exp(-u)$ is also of class $C^{2+\alpha}$ up to the boundary. The proof relies on new Schauder estimates for degenerate elliptic equations of Fuchsian type.