Convexity for free boundaries with singular term (nonlinear elliptic case)

TL;DR

This paper investigates a nonlinear elliptic free boundary problem with a convex fixed boundary, proving the existence of quasi-concave solutions under certain conditions, and extends results to specific geometric configurations.

Contribution

It establishes the existence of quasi-concave solutions for a class of nonlinear elliptic free boundary problems with convex fixed boundaries, including cases with specific geometric constraints.

Findings

Existence of nonnegative quasi-concave solutions proven.

Results extend to cases where the fixed boundary set is contained in a hyperplane.

Applicable to fully nonlinear and p-Laplace operators.

Abstract

We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & \text{in }\Omega\setminus K, \\ u=1 & \text{on }\partial K,\\ |\nabla u|=0 &\text{on }\partial \Omega, \end{array}\end{cases} where $K$ is a (given) convex and compact set in $\mathbb{R}^n$ ($n\ge2$), $\Omega=\{u>0\}\supset K$ is an unknown set, and $L$ is either a fully nonlinear or the $p$-Laplace operator. Under suitable assumptions on $K$ and $g$, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set $K$ is contained in $\{x_n=0\}$, and obtain similar results.