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Free discontinuity problems with unbounded data: The two dimensional case. (English) Zbl 0774.49009
We prove the existence of a minimizing pair for the functional \(\mathcal G\) defined for every closed set \(K\subset{\mathbf R}^ 2\) and for every function \(u\in C^ 1(\Omega\backslash K)\) by \[ {\mathcal G}(K,u)=\int_{\Omega\backslash K} |\nabla u|^ 2 d{\mathcal L}^ 2+\mu \int_{\Omega\backslash K} | u-g|^ q d{\mathcal L}^ 2+\lambda{\mathcal H}^ 1(K\cap \Omega), \] where \(\Omega\) is an open set in \({\mathbf R}^ 2\), \(\lambda,\mu>0\), \(q\geq 1\), \(g\in L^ q(\Omega)\cap L^ p(\Omega)\) with \(p>2q\), \({\mathcal L}^ 2\) is the Lebesgue measure and \({\mathcal H}^ 1\) is the 1-dimensional Hausdorff measure. We show that a minimizing pair for \(\mathcal G\) does not exist for a suitable \(g\in L^ p(\Omega)\cap L^ q(\Omega)\) for every \(p<2q\). The existence result has been improved with \(p=2q\) and extended to the \(n\)-dimensional case with \(p\geq nq\) in a subsequent paper.
The functional \(\mathcal G\) has been considered (with \(q=2\) and \(g\in L^ \infty(\Omega))\) by D. Mumford and J. Shah in the framework of image segmentation in Computer Vision Theory. For further applications see E. De Giorgi [in: Frontiers in Pure and Applied Mathematics, 55-62 (1991; Zbl 0758.49002)].
Reviewer: A.Leaci (Lecce)

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
68U10 Computing methodologies for image processing
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