Estimates for minimizers of some relaxed polyconvex functionals. (August 2021)

• Record Type: Journal Article
• Authors: Carozza, Menita; Esposito, Luca; Leonetti, Francesco
• Is Part Of: Nonlinear analysis. Volume 209(2021)
• Publication Date: 2021-08
• Subjects: 49N99 -- 35B50; Polyconvex -- Functional -- Relaxation -- Minimizer -- Regularity; 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.2021.112334
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the relaxed functional RF ( u ) = inf lim inf k F ( u k ) : u k → u where F is the polyconvex integral F ( u ) = ∫ Ω [ | D u | p + h ( det D u ) ] d x, with u : Ω ⊂ R n → R n and h ≥ 0 is convex. We prove bounds for minimizers of RF ( u ) . Similar results are already known when p ≥ 2 . In the present paper we use a different technique that allows us to get also the subquadratic case 1 < p < 2 . The model case is h ( t ) = | t | s with s ≥ 1 : with such an h, we get maximum modulus inequality sup Ω | u | ≤ sup ∂ Ω | u | .