The Dirichlet problem for the $\alpha$-singular minimal surface equation

TL;DR

This paper proves the existence and uniqueness of classical solutions for the Dirichlet problem associated with the $ ext{alpha}$-singular minimal surface equation in mean convex domains when $ ext{alpha}<0$, for arbitrary continuous boundary data.

Contribution

It establishes the first existence and uniqueness results for the $ ext{alpha}$-singular minimal surface equation with arbitrary boundary data in mean convex domains for negative $ ext{alpha}$.

Findings

Existence and uniqueness of solutions for $ ext{alpha}<0$

Solutions hold for arbitrary continuous boundary data

Results apply to bounded mean convex domains

Abstract

Let $\Omega\subset\r^n$ be a bounded mean convex domain. If $\alpha<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $\Omega$ for the $\alpha$-singular minimal surface equation with arbitrary continuous boundary data.