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

Let Ω 2 be a bounded piecewise C1,1 open set with convex corners, and let

MS(u) :=Ω|u|2dx + α1(J u) + βΩ|u g|2dx

be the Mumford–Shah functional on the space SBV(Ω), where g L(Ω) and α,β > 0. We prove that the function u H1(Ω) such that

Δu + βu = βg in Ω, uν = 0  on  Ω

is a local minimizer of MS with respect to the L1-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.

local minimality, monotonicity formulas, free discontinuity functionals
Mathematical Subject Classification 2010
Primary: 35R35, 35A16, 28A75, 35J25, 35Q74, 49J45, 94A08
Received: 16 October 2018
Revised: 9 February 2019
Accepted: 3 April 2019
Published: 15 April 2020
Dorin Bucur
Université Savoie Mont Blanc
Ilaria Fragalà
Dipartimento di Matematica
Politecnico di Milano
Alessandro Giacomini
DICATAM, Sezione di Matematica
Università degli studi di Brescia