A discrete Bernoulli free boundary problem

TL;DR

This paper introduces a new formulation of the Bernoulli free boundary problem involving the p-Laplace operator, replacing boundary gradient conditions with level surface distance conditions, simplifying analysis and enabling new theoretical insights.

Contribution

It proposes a novel free boundary problem formulation related to the Bernoulli problem, with convergence results and existence and qualitative theory in convex and other regimes.

Findings

Model converges to classical Bernoulli problem under suitable scalings

Simplifies analysis by avoiding boundary gradient considerations

Establishes existence and qualitative properties in convex regimes

Abstract

We consider a free boundary problem for the $p$-Laplace operator which is related to the so-called Bernoulli free boundary problem. In this formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. For suitable scalings our model converges to the classical Bernoulli problem; one of the advantages in this new formulation lies in the simplicity of the arguments, since one does not need to consider the boundary gradient. We shall study this problem in convex and other regimes, and establish existence and qualitative theory.