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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
##### Keywords:
minimization; regularity; Mumford-Shah functional
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##### References:
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