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 a functional introduced in a classical paper of Alt and Caffarelli. 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.
A second order minimality condition for a free-boundary problem
Mora, M. G.
2019-01-01
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 a functional introduced in a classical paper of Alt and Caffarelli. 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.File in questo prodotto:
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