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Existence theory for a new class of variational problems. (English) Zbl 0711.49064
The paper deals with the existence of minimizers of functionals of the form \[ \int_{\Omega \setminus \Gamma}f(x,u,\nabla u)dx+\int_{\Gamma}\phi (x,u^+,u^-,\nu)d{\mathcal H}^{n-1} \] whese \(\Omega \subset {\mathbb{R}}^ n\) and \({\mathcal H}^{n-1}\) is the Hausdorff (n- 1)-dimensional measure. The function u: \(\Omega\to {\mathbb{R}}^ k\) need not be continuous: the surface energy is obtained by integrating on the discontinuity set \(\Gamma\) of u an energy density \(\phi\) depending on x, the normal \(\nu\) to \(\Gamma\) and the asymptotic value \(u^+,u^-\) of u near x. The appropriate function space of the functional is a suitable subset \(SBV(\Omega,{\mathbb{R}}^ k)\) of vector valued functions of bounded variation. General compactness and lower semicontinuity results are established.
Reviewer: L.Ambrosio

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
28A75 Length, area, volume, other geometric measure theory
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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