A second order minimality condition for the Mumford-Shah functional

Details A second order minimality condition for the Mumford-Shah functional A second order minimality condition for the Mumford-Shah functional

2007-04-24

arxiv.org/abs/0704.3124v1

Mathematics

Analysis of PDEs

30 pages

Scientific paper

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

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