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Functionals depending on curvatures with constraints. (English) Zbl 1017.49019

In this paper, the authors consider functionals of the form \[ F(E)=\int_{\partial E} \varphi(K_1, \ldots, K_{n-1}) d{\mathcal H}^{n-1}, \] where \(\varphi\) is a convex function on \({\mathbf R}^{n-1}\), \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure, and \(K_i\) are the elementary symmetric functions of the principal curvatures of the boundary \(\partial E\) of a sufficiently regular closed set \(E\subset\mathbf R^n\). For \(R>0\) they define the set \(\mathcal U_R\) of closed bounded sets in \(\mathbf R^n\) such that, for any boundary point \(p\in\partial E\) there are two open balls of radius \(R\) tangent at \(p\), one of them contained in \(E\) and the other one in the exterior of \(E\).
The authors show that the sets in class \(\mathcal U_R\) are sufficiently regular to allow the functional \(F\) to be defined on them. Next, they show compactness and semicontinuity results for \(F\) in \(\mathcal U_R\). For the compactness result the topology on \(\mathcal U_R\) is the one determined by the convergence in the sense of Kuratowski, which is induced on the space of equibounded sets by the Hausdorff distance. For the semicontinuity result the convergence on \(\mathcal U_R\) is the \(L^1\)-convergence of characteristic functions.
They apply their results in \(\mathbf R^2\) to show existence of solutions in \(\mathcal U_R\) to a variational problem, proposed by M. Nitzberg and D. Mumford, related to image segmentation models.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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