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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.

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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