Calculus of Variations and Geometric Measure Theory

I. Fonseca - G. Leoni - M. G. Mora

A Second Order Minimality Condition for a Free-Boundary Problem

created by mora on 26 May 2016
modified on 05 Jun 2017


Submitted Paper

Inserted: 26 may 2016
Last Updated: 5 jun 2017

Year: 2016


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 \[ v\mapsto\int_{\Omega}\big( \lvert \nabla v\rvert ^{2}+\chi_{\{v>0\}}Q^{2}% \big)\,d\boldsymbol{x}, \] 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.