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Regularity properties of free discontinuity sets. (English) Zbl 1024.49013
In this paper the authors have studied the regularity properties of a singular set \(K\) minimizing the Mumford-Shah functional. They deal with the problem of establishing that the set of points where \(K\) is not locally a \(C^1\) surface is “small”. More exactly, they have proved that every ball centered on \(K\) contains, at an estimable scale transition, a subball in which \(K\) is a \(C^1\) surface splitting the ball in two hemispheres on which the oscillation of the function \(u_K\) is small, where \((K,u_K)\) is the pair which minimizes the Mumford-Shah functional.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control
49J10 Existence theories for free problems in two or more independent variables
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