Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity

Abstract

We consider families of variational problems F ε over domains Ω ε whose extension in one or more directions is small compared to the extension in the other directions, and goes to zero while ε tends to zero. We study then the “variational” convergence of the functionals F ε to a new functional defined on a domain A in a lower dimensional space, where those “dimensions” that were small in Ω ε disappear. A general framework is presented in the first part of the paper and an application to the elastic rod and the elastic plate is given in the second part.