On Problems in the Calculus of Variations in Increasingly Elongated Domains

Abstract

Laboratoire d'équations aux dérivées partielles non linéaires et histoire des mathématiques, ENS, B.P. 92, Vieux Kouba, 16050 Alger, Algérie and USTHB, Faculté des mathématiques, Département d'analyse, Laboratoire d'analyse mathématique et numérique des équations aux dérivées partielles, Bab Ezzouar, Alger, Algérie. AbstractWe consider minimization problems in the calculus of variations set in a sequence of domains the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. We study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions. MSC 2010: 35J25, 35J35, 35J62, 35J92, 49J45, 74K99. Keywords: Calculus of variations, domains becoming unbounded, asymptotic behavior, exponential rate of convergence.