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 \(1<p<\infty\). 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 \(\mathbf{R}^n \setminus \{0\}\). We also show that the super level sets \(\{u>t\}\) are convex for \(t\in (0,1)\).