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Relaxation results for some free discontinuity problems. (English) Zbl 0817.49015
In many problems of Mathematical Physics one is concerned with minimum problems for functions defined on function spaces allowing discontinuities. When the discontinuity set is not a priori given the problem is called with “free discontinuity” and it would be helpful, for the existence theory and for the study of minimizing sequences, the knowledge of lower semicontinuity theorems and of the related relaxation.
This kind of problem has been recently introduced by E. De Giorgi and L. Ambrosio [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82, No. 2, 199-210 (1988; Zbl 0715.49014)], where the existence of minimizers is proved via the direct method of the calculus of variations. Here the authors consider functionals of the form \[ F(u)= \int_ \Omega f(| \nabla u|) dx+ \int_{S(u)} g(| [u]|) dH^{n- 1} \] defined on the space \(\text{SBV}(\Omega; \mathbb{R}^ m)\) and, under suitable conditions on \(f\) and \(\varphi\), they identify the relaxed functional as an integral on the whole space \(\text{BV}(\Omega; \mathbb{R}^ m)\) of the form \[ \overline F(u)= \int_ \Omega \overline f(|\nabla u|) dx+ c| C(u)| (\Omega)+ \int_{S(u)} \overline g(|[u]|) dH^{n-1}, \] where \(Du= \nabla u dx+ [u]\otimes \nu dH^{n-1}+ C(u)\) is the decomposition of the measure \(Du\) into absolute continuous, jump, and Cantor parts.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49M20 Numerical methods of relaxation type
49S05 Variational principles of physics
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