Calculus of Variations and Geometric Measure Theory

M. G. Mora - M. Morini

Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set

created on 14 Dec 2001
modified on 17 Mar 2003


Published Paper

Inserted: 14 dec 2001
Last Updated: 17 mar 2003

Journal: Ann. Inst. H. Poincaré Anal. Nonlin.
Volume: 18
Pages: 403-436
Year: 2001


Using a calibration method, we prove that, if $w$ is a function which satisfies all Euler conditions for the Mumford-Shah functional on a two-dimensional open set $\Omega$, and the discontinuity set $S_w$ of $w$ is a regular curve connecting two boundary points, then there exists a uniform neighbourhood $U$ of $S_w$ such that $w$ is a minimizer of the Mumford-Shah functional on $U$ with respect to its own boundary conditions on $\partial U$. We show that Euler conditions do not guarantee in general the minimality of $w$ in the class of functions with the same boundary value of $w$ on $\partial \Omega$ and whose extended graph is contained in a neighbourhood of the extended graph of $w$, and we give a sufficient condition in terms of the geometrical properties of $\Omega$ and $S_w$ under which this kind of minimality holds.

Keywords: Mumford-Shah functional, calibration method