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A derivation formula for convex integral functionals defined on $$\text{BV} (\Omega)$$. (English) Zbl 0842.49015
After recalling definitions and some facts concerning the functions of bounded variation and sets of finite perimeter the authors prove a derivation formula for positively one-homogeneous functionals on the space $$\text{BV}(\Omega)$$ (the space of all scalar-valued functions of bounded variation), where $$\Omega$$ is an open set in $$\mathbb{R}^n$$. This theorem can be thought of as a representation result for such functionals defined on the Sobolev space $$W^{1,1}(\Omega)$$ and of functionals defined on “partitions of $$\Omega$$ in sets of finite perimeter”. An application to homogenization of positively one-homogeneous functionals is also given. Similar results but for functionals defined on Sobolev spaces were studied by G. Bouchitté, D. Dal Maso, L. Modica, and others.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting 49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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