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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
