Vol. 13, No. 3, 2020

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Local minimality results for the Mumford–Shah functional via monotonicity

Dorin Bucur, Ilaria Fragalà and Alessandro Giacomini

Vol. 13 (2020), No. 3, 865–899
Abstract

Let $\Omega \subseteq {ℝ}^{2}$ be a bounded piecewise ${C}^{1,1}$ open set with convex corners, and let

$MS\left(u\right):={\int }_{\Omega }|\nabla u{|}^{2}\phantom{\rule{0.3em}{0ex}}dx+\alpha {\mathsc{ℋ}}^{1}\left({J}_{u}\right)+\beta {\int }_{\Omega }|u-g{|}^{2}\phantom{\rule{0.3em}{0ex}}dx$

be the Mumford–Shah functional on the space $SBV\left(\Omega \right)$, where $g\in {L}^{\infty }\left(\Omega \right)$ and $\alpha ,\beta >0$. We prove that the function $u\in {H}^{1}\left(\Omega \right)$ such that

is a local minimizer of $MS$ with respect to the ${L}^{1}$-topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasiminimizers of the Mumford–Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.

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