Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals. (December 2022)

• Record Type: Journal Article
• Authors: Kreisbeck, Carolin; Ritorto, Antonella; Zappale, Elvira
• Is Part Of: Nonlinear analysis. Volume 225(2022)
• Publication Date: 2022-12
• Subjects: primary 49J45 26B25; Nonlocality -- Supremal functionals -- Double integrals -- Relaxation -- Lower semicontinuity -- γ-convergence -- Lp-approximation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113111
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Motivated by the direct method in the calculus of variations in L ∞, our main result identifies the notion of convexity characterizing the weakly ∗ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in Kreisbeck and Zappale (2020), but also establishes a connection with double integrals. We relate the two classes of functionals via an L p -approximation in the sense of Γ -convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.