×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: EMIS EuDML