Estimates for minimizers of some relaxed polyconvex functionals. (August 2021) - Carozza, Menita; Esposito, Luca; Leonetti, Francesco
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 | .