The -convergence of oscillating integrands with nonstandard coercivity and growth conditions

Abstract

We study the -convergence as of a family of integral functionals with integrand , where the integrand oscillates with respect to the space variable . The integrands satisfy a two-sided power estimate on the coercivity and growth with different exponents. As a consequence, at least two different variational Dirichlet problems can be connected with the same functional. This phenomenon is called Lavrent'ev's effect. We introduce two versions of -convergence corresponding to variational problems of the first and second kind. We find the -limit for the aforementioned family of functionals for problems of both kinds; these may be different. We prove that the -convergence of functionals goes along with the convergence of the energies and minimizers of the variational problems. Bibliography: 23 titles.