Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Abstract

Abstract. New L 1 -lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a representation formula for the relaxed functional is also obtained.Mathematics Subject Classification. 49J45, 49Q20, 49M20.

The Dirichlet problem associated to the relativistic heat equation

“…Note that this functional verifies the hypothesis of Theorem 3.1 in [1] and then, it is lower semi-continuous with respect to the L 1 -convergence. This implies the lower semi-continuity of the functional F g φl (u) since…”

“…Equation (1) was derived by Brenier [19] by means of Monge-Kantorovich's mass transport theory and he named it as the relativistic heat equation.…”

Relaxation of signed integral functionals in BV

“…in by the Besicovitch Differentiation Theorem (cf. (5)) and Alberti's Rank One Theorem. As we go along, we will require more of x 0 , but every time the discarded set is |D s u|-negligible.…”

“…The BV-case was then solved by Ambrosio and Dal Maso [6] as well as by Fonseca and Müller [20], both results being based on the blow-up technique. The paper [20] also contained results for x-and u(x)-dependent integrands and there is an ongoing effort to generalize the conditions required on f , see for example [4,5,14,27].…”

Weak lower semicontinuity for non coercive polyconvex integrals

“…Some extensions of Serrin's results have been recently obtained by and by several other authors, also in the context of BV -functions (see [18,15,9,10,3]). …”