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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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