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Integral representation results for functionals defined on $$\text{SBV}(\Omega;\mathbb{R}^ m)$$. (English) Zbl 0880.49010
The main result of the paper concerns the integral representation (with Carathéodory integrands) of functionals of the form $F: \text{SBV}(\Omega;{\mathbb{R}}^m) \times {\mathcal B}(\Omega) \longrightarrow [0,+\infty],$ satisfying some properties like “locality”, “lower semicontinuity”, “growth conditions”, and so on…. Above $$\text{SBV}(\Omega,{\mathbb{R}}^m)$$ denotes the Ambrosio-De Giorgi space of special functions of bounded variation defined on an open set $$\Omega \in {\mathbb{R}}^n$$ and with values in $${\mathbb{R}}^m$$, while $${\mathcal B}(\Omega)$$ stands for the family of Borel subsets of $${\Omega}$$. The theorem presented in the paper generalizes an analogous result of Buttazzo and Dal Maso proved for functionals defined on Sobolev spaces. The proof of the theorem involves blow-up techniques and approximation results for functions of bounded variation. Some applications to the relaxation in image segmentation problems, as well as in fracture mechanics (more precisely to the relaxation of nonlinear elasticity energies for brittle fracture problem) are also given.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 74B20 Nonlinear elasticity 74R99 Fracture and damage 49J10 Existence theories for free problems in two or more independent variables