Existence theory for a new class of variational problems (Q923364)

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.