On the Lower Semicontinuity of Supremal Functionals

Abstract

Abstract. In this paper we study the lower semicontinuity problem for a supremal functional of the Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogous theory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.Mathematics Subject Classification. 49J45, 49L25.

Γ-convergence and absolute minimizers for supremal functionals

“…• in [18] the authors consider a Serrin-type semicontinuity theorem (i.e. without coercivity assumptions) in the scalar case for the functional (3.3).…”

On supremal functionals defined in BV

“…Moreover, by means of this result, it was assured the existence of a minimum point for a suitable generalized Dirichlet problem defined on BV( ) even without coercivity assumptions of f (see [17,Theorem 11]). In the present paper, just moving from the quoted results, we approach the more general problem of the lower semicontinuity of supremal functional on BV( ) associated to functions f (x, t, ξ) depending also on the geometric variable x and on the function variable t. Following the work of Dal Maso [7], we propose here a supremal functional defined on BV( ), very similar to (1.1), given by…”

“…Very recently, starting from these considerations, the problem of the definition and the lower semicontinuity of functionals as (1.2) on BV( ) has been approached (see [1,17,18]) and in particular it was proved that the lower semicontinuity and the level convexity of f : R N → [0, ∞] imply the lower semicontinuity on BV( ) with respect to the w * − BV( ) convergence of the functional…”

“…where D s u = D c u + D j u and f is the so-called asymptotic function of f (see [17,Theorem 10]]). Moreover, by means of this result, it was assured the existence of a minimum point for a suitable generalized Dirichlet problem defined on BV( ) even without coercivity assumptions of f (see [17,Theorem 11]).…”

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

“…The model case where f (x, u, ∇u) ≡ |∇u| is related to the classical problem of finding the best Lipschitz constant of a function with prescribed boundary data, first considered by McShane in [18]. In order to apply the direct method of the calculus of variations the main issue is the lower semicontinuity of F. Semicontinuity properties for supremal functionals have been studied by many authors in the last years; we refer for instance to , Barron-Liu [5], Barron-Jensen-Wang [4], and to the recent papers by Prinari [19] and Gori-Maggi [17]. In [4] the authors proved a lower semicontinuity result for Funder the assumption (called level convexity) that the sub levels of f (x, u, ·) are convex.…”