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In this paper it is shown that any regular critical point of the Mumford–Shah functional, with positive definite second variation, is an isolated local minimizer with respect to competitors which are sufficiently close in the L 1 -topology. A global minimality result in small tubular neighborhoods of the discontinuity set is also established. [ABSTRACT FROM AUTHOR]

Abstract: We consider a conducting body which presents some (unknown) perfectly insulating defects, such as cracks or cavities, for instance. We aim to reconstruct the defects by performing measurements of current and voltage type on a (known and accessible) part of the boundary of the conductor. A crucial step in this reconstruction is the determination of the electrostatic potential inside the conductor, by the electrostatic boundary measurements performed. Since the defects are unknown, we state such a determination problem as a free-discontinuity problem for the electrostatic potential in the framework of special functions of bounded variation. We provide a characterisation of the looked for electrostatic potential and we approximate it with the minimum points of a sequence of functionals, which take also in account the error in the measurements. These functionals are related to the so-called Mumford–Shah functional, which acts as a regularizing term and allows us to prove existence of minimizers and Γ-convergence properties. [Copyright &y& Elsevier]

We study by means of $\Gamma$-convergence the asymptotics of the rescaled Mumford-Shah functional when $\varepsilon \to 0$ and prove the existence of a $\Gamma$-limit. The limit functional is easy to handle and can be used as a simple approximation to the original Mumford-Shah functional. Moreover, its minimizers can be interpreted as a sort of asymptotic probability distribution of the sets $\Gamma$. Some examples illustrate the use of this method in image segmentation. [ABSTRACT FROM AUTHOR]

In this paper we give a partial answer to a conjecture of De Giorgi, namely we prove that in dimension two the regular part of the discontinuity set of a local minimizer of the homogeneous Mumford–Shah functional is analytic with the exception of at most a countable number of isolated points. [Copyright &y& Elsevier]