On a Bernoulli-type overdetermined free boundary problem

Murat Akman¹ ; Agnid Banerjee² ; Mariana Smit Vega Garcia³

¹ University of Essex, Department of Mathematical Sciences
² Tata Institute of Fundamental Research, Center for Applicable Mathematics
³ Western Washington University, Department of Mathematics

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 601-618.

Résumé

  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 $t\in (0,1)$.

Détail

Keywords: Quasilinear elliptic equations and p-Laplacian, degenerate elliptic equations, free boundary problems, Bernoulli-type free boundary problems