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 [26] to A-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation for a fixed 1 < p < ∞. 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 Ω with K ⊂ Ω and a function u which is A-harmonic in Ω \ K, has continuous boundary values 1 on ∂K and 0 on ∂Ω, such that |∇u| = c on ∂Ω. Moreover, ∂Ω is C 1,γ for some γ > 0, and it is smooth provided A is smooth in R n \ {0}. We also show that the super level sets {u > t} are convex for t ∈ (0, 1).