The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions
Vasilii Vasil'evich Zhikov, Svetlana Evgenievna Pastukhova · Sbornik Mathematics · 2014
We study the Γ-convergence as e→0 of a family of integral functionals with integrand f{sub e}(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 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 Γ-convergence corresponding to variational 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. Bibliography: 23 titles. (paper)