Lipschitz minimizers for a class of integral functionals under the bounded slope condition. (March 2022)

• Record Type: Journal Article
• Authors: Don, Sebastiano; Lussardi, Luca; Pinamonti, Andrea; Treu, Giulia
• Is Part Of: Nonlinear analysis. Volume 216(2022)
• Publication Date: 2022-03
• Subjects: 49J45 -- 26A45 -- 49Q05 -- 53C17; Minimization problem -- Relaxation -- Bounded slope condition -- Functions of bounded variation -- Heisenberg groups; 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.2021.112689
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the functional ∫ Ω g ( ∇ u + X ∗ ) d L 2 n where g is convex and X ∗ ( x, y ) = 2 ( − y, x ) and we study the minimizers in BV ( Ω ) of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on g, there exists a unique minimizer which is also Lipschitz continuous. The assumptions on g allow to consider both the case with superlinear growth and the one with linear growth. Moreover neither uniform ellipticity nor smoothness of g are assumed.