# Science at the triple point between mathematics, mechanics and materials science

## Publication 125

### A Second Order Minimality Condition for a Free-Boundary Problem

##### Authors:

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Maria Giovanna Mora
Dipartimento di Matematica
Universita di Pavia
Pavia, Italy

##### 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 $$v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2}% \big)\,d\boldsymbol{x},$$ 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.
##### Get the paper in its entirety
16-CNA-010.pdf

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