Existence theory for a new class of variational problems. (English) Zbl 0711.49064

Ambrosio, L.

Arch. Ration. Mech. Anal. 111, No. 4, 291-322 (1990).

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

MSC:

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

Keywords:

existence of minimizers; compactness; lower semicontinuity

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