A second order minimality condition for a free-boundary problem

TL;DR

This paper establishes necessary and sufficient second variation conditions for minimality in a two-dimensional free-boundary problem, extending classical results and including applications to water waves.

Contribution

It derives a complete second order minimality condition for a classical free-boundary functional, linking second variation positivity to local minimality in 2D.

Findings

Second variation positivity implies local minimality.

Smooth critical points are local minimizers nearby.

Results apply to water wave models.

Abstract

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2} \big)\,dx, \] introduced in a classical paper of Alt and Caffarelli. For a special choice of $Q$ this includes water waves. The second variation is obtained by computing the second derivative of the functional along suitable variations of the free boundary. It is proved that the strict positivity of the second variation gives a sufficient condition for local minimality. Also, it is shown that smooth critical points are local minimizers in a small tubular neighborhood of the free-boundary.