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The calibration method for the Mumford-Shah functional and free-discontinuity problems. (English) Zbl 1015.49008

The Mumford-Shah functional defined as \[ F(u)=\int_{\Omega\backslash S_u}|\nabla u|^2 +\alpha{\mathcal H}^{n-1}(S_u)+\beta\int_\Omega(u-g)^2 , \] was introduced within the context of a variational approach to image segmentation problems, and can be regarded as the prototypical example of functional coupling bulk and surface contributions. In it \(\Omega\) is a bounded regular domain in \({\mathbb R}^n\), \(g: \Omega\to[0,1]\) is a given function, \(\alpha\), \(\beta\geq 0\), and \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure.
One of the most relevant features of \(F\) is a deep lack of convexity. So, even if suitable equilibrium conditions can be obtained by considering different types of infinitesimal variations, in general they do not imply minimality.
In the paper a sufficient condition for minimality is proposed that closely resembles the classical principle of calibrations for minimal surfaces, and some applications of it are described.
Albeit the Mumford-Shah functional cannot be included in the general theory of calibrations, some examples of calibration for \(F\) are proposed, and short and easy proofs of several minimality results are given.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49K10 Optimality conditions for free problems in two or more independent variables
49Q20 Variational problems in a geometric measure-theoretic setting
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