The interaction between bulk energy and surface energy in multiple integrals
Andrea Braides, Alessandra Coscia · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1994
This paper is devoted to the study of integral functional denned on the space SBV (Ω ℝ k ) of vector-valued special functions with bounded variation on the open set Ω⊂ℝ n , of the form We suppose only that f is finite at one point, and that g is positively 1-homogeneous and locally bounded on the sets ℝ k ⊗ v m , where { v 1 ,…, v R } ⊂ S n−1 is a basis of ℝ n . We prove that the lower semicontinuous envelope of F in the L 1 (Ω;ℝ k )-topology is finite and with linear growth on the whole BV (Ω;ℝ k ), and that it admits the integral representation A formula for ϕ is given, which takes into account the interaction between the bulk energy density f and the surface energy density g .