Using a calibration method, we prove that, if w is a function that satisfies all the 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. We show that the Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w 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.

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

Mora, M. G.;
2001-01-01

Abstract

Using a calibration method, we prove that, if w is a function that satisfies all the 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. We show that the Euler conditions do not guarantee in general the minimality of w in the class of functions with the same boundary value of w 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/363562
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