Analyticity of the solutions to degenerate Monge-Amp\`ere equations

TL;DR

This paper proves that solutions to a class of degenerate Monge-Ampère equations are analytic within the domain when the exponent q is a positive integer, under conditions of convexity and analyticity of the domain.

Contribution

It establishes the analyticity of solutions to degenerate Monge-Ampère equations for integer q, extending regularity results to degenerate cases.

Findings

Solutions are analytic in the closure of the domain for positive integer q.

Analyticity holds under the assumptions of uniform convexity and analyticity of the domain.

The result applies specifically to the case where q is a positive integer.

Abstract

This paper is devoted to study the following degenerate Monge-Amp\`ere equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=\Lambda_q (-u)^q \quad \text{in}\quad \Omega,\\ u=0 \quad\text{on}\quad \partial\Omega \end{cases} \end{eqnarray} for some positive constant $\Lambda_q$. Suppose $\Omega\subset\subset \mathbb R^n$ is uniformly convex and analytic. Then the solution of the degenerate Monge-Amp\`ere equation is analytic in $\bar\Omega$ provided $q\in \mathbb Z^+$.