Concavity properties for free boundary elliptic problems

TL;DR

This paper investigates concavity properties of free boundary problems, establishing inequalities and uniqueness results that deepen understanding of the geometric and analytical aspects of Bernoulli-type problems.

Contribution

It introduces new concavity inequalities and a uniqueness theorem for nonlinear Bernoulli free boundary problems, expanding theoretical understanding.

Findings

Proved a Brunn-Minkowski inequality for the Bernoulli Constant.

Established an Urysohn's type inequality for the Bernoulli Constant.

Demonstrated a uniqueness result for the interior nonlinear Bernoulli problem.

Abstract

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem.