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arXiv · 0704.3124

A second order minimality condition for the Mumford-Shah functional

Filippo Cagnetti·Maria Giovanna Mora·Massimiliano Morini
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Abstract

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 $H^1_0(Γ)$, $Γ$ 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 $Γ$. 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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Filippo Cagnetti, Maria Giovanna Mora, Massimiliano Morini. 2007-04-24. A second order minimality condition for the Mumford-Shah functional. https://arxiv.org/abs/0704.3124

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