Local minimality results for the Mumford–Shah functional via monotonicity

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

\{\begin{matrix} - Δ u + β u = β g & in Ω, \\ \partial u ∕ \partial ν = 0 & on \partial Ω \end{matrix}

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.

Keywords

local minimality, monotonicity formulas, free discontinuity functionals

Mathematical Subject Classification 2010

Primary: 35R35, 35A16, 28A75, 35J25, 35Q74, 49J45, 94A08

Milestones

Received: 16 October 2018

Revised: 9 February 2019

Accepted: 3 April 2019

Published: 15 April 2020

Authors

Dorin Bucur
Université Savoie Mont Blanc CNRS, LAMA Chambéry France

Ilaria Fragalà
Dipartimento di Matematica Politecnico di Milano Milano Italy

Alessandro Giacomini
DICATAM, Sezione di Matematica Università degli studi di Brescia Brescia Italy

Vol. 13, No. 3, 2020

Dorin Bucur, Ilaria Fragalà and Alessandro Giacomini

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors