A Bernoulli problem with non constant gradient boundary constraint

TL;DR

This paper investigates a Bernoulli free boundary problem with a non-constant gradient boundary condition depending on the outer normal, establishing existence and convexity of solutions under certain conditions.

Contribution

It proves that in the convex case, the existence of a subsolution ensures a convex classical solution for this specific Bernoulli problem.

Findings

Existence of solutions is guaranteed by subsolutions in convex cases.

Solutions are proven to be convex.

The boundary constraint depends on the outer unit normal.

Abstract

We present in this paper a result about existence and convexity of solutions to a free boundary problem of Bernoulli type, with non constant gradient boundary constraint depending on the outer unit normal. In particular we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.