Publication 16-CNA-010
A Second Order Minimality Condition for a
Free-Boundary Problem
Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu
Giovanni Leoni
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
giovanni@andrew.cmu.edu
Maria Giovanna Mora
Dipartimento di Matematica
Universita di Pavia
Pavia, Italy
mariagiovanna.mora@unipv.it
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 as 16-CNA-010.pdf
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