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

摘要

We study the Γ-convergence as ε → 0 of a family of integral functionals with integrand f_ε(x, u, ▽u), where the integrand oscillates with respect to the space variable x. The integrands satisfy a two-sided power estimate on the coercivity and growth with different exponents. As a con- sequence, at least two different variational Dirichlet problems can be con- nected with the same functional. This phenomenon is called Lavrent'ev's effect. We introduce two versions of Γ-convergence corresponding to vari- ational problems of the first and second kind. We find the Γ-limit for the aforementioned family of functionals for problems of both kinds; these may be different. We prove that the Γ-convergence of functionals goes along with the convergence of the energies and minimizers of the variational problems.