On the existence of solutions to a problem in multidimensional segmentation. (English) Zbl 0729.49003

The authors give an existence result for a minimum problem related to image segmentation in computer vision. In the framework of the calculus of variations this minimum problem pertains to the class of “Minimal boundary problems with free discontinuity” recently studied by De Giorgi-Carriero and Leaci. The main result is the following:
Let \(n\geq 2\), M open \(\subset R^ n\), \(0<\lambda <+\infty\), \(1\leq p<+\infty\), \(g\in L^ p(M)\cap L^{\infty}(M)\). Then there exists at least one pair (K,u) minimizing \[ F(K,u)=\lambda \int_{M\setminus K}| u-g|^ p dx+H^{n-1}(K\cap M), \] with k closed \(\subset R^ n\), \(u\in C'(M\setminus K)\) such that \(\nabla u\equiv 0\) in \(M\setminus K\).


49J10 Existence theories for free problems in two or more independent variables
49N70 Differential games and control
49N75 Pursuit and evasion games
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