On a Bernoulli-type overdetermined free boundary problem

Abstract

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in [25] to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the $p$-Laplace equation for a fixed $1<p<\mathrm{\infty }$. 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 $\mathrm{\Omega }$ with $K\subset \mathrm{\Omega }$ and a function $u$ which is $\mathcal{A}$-harmonic in $\mathrm{\Omega }\setminus K$, has continuous boundary values 1 on $\partial K$ and 0 on $\partial \mathrm{\Omega }$, such that $|\mathrm{\nabla }u|=c$ on $\partial \mathrm{\Omega }$. Moreover, $\partial \mathrm{\Omega }$ is $C^{1,\gamma }$ for some $\gamma >0$, and it is smooth provided $\mathcal{A}$ is smooth in ${\mathbf{R}}^n\setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$