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

✍ Scribed by Maria Giovanna Mora; Massimiliano Morini

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 360 KB
• Volume: 18
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(01)00075-0

✦ Synopsis

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 , 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 ∂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 ∂ 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 and S w under which this kind of minimality holds.  2001 Éditions scientifiques et médicales Elsevier SAS