On a Bernoulli-type overdetermined free boundary problem

TL;DR

This paper extends the study of Bernoulli-type free boundary problems to a-harmonic PDEs, proving existence, uniqueness, regularity, and convexity of solutions and free boundaries in a generalized setting.

Contribution

It generalizes previous work to a-harmonic PDEs, establishing existence, uniqueness, regularity, and convexity results for the free boundary problem.

Findings

Existence and uniqueness of convex solutions with prescribed boundary conditions.

Regularity results showing a-harmonic free boundaries are $C^{1,eta}$ and smooth under certain conditions.

Convexity of super level sets of the solution.

Abstract

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the interior ball condition and $c>0$ is a given constant, then there exists a unique convex domain $\Omega$ with $K\subset \Omega$ and a function $u$ which is $\mathcal{A}$-harmonic in $\Omega\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on $\partial\Omega$, such that $|\nabla u|=c$ on $\partial \Omega$. Moreover, $\partial\Omega$ is $C^{1,\gamma}$ for some $\gamma>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n \setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$.