$ \Gamma$-convergence of functionals with periodic integrands via $ 2$-scale convergence

M. Baía and I. Fonseca

Abstract:

$ \Gamma$-convergence techniques combined with techniques of $ 2$-scale convergence are used to give a characterization of the behavior as $ \epsilon$ goes to zero of a family of integral functionals defined on $ L^{p}(\Omega;\\ mathbb{R}^{d})$ by

$\displaystyle {\cal I}_{\epsilon}(u):=\left\{\begin{array}{l} \int_{\Omega} f\l... ...mathbb{R}^{d}), \\ \\ +\infty \hspace{3cm}\mbox{otherwise}, \end{array}\right.$

under periodicity (and nonconvexity) hypothesis, standard $ p$-coercivity and $ p$-growth conditions with $ p>1$. Precisely it is assumed that the integrand $ f=f(x,y,\xi)$ is measurable in $ x$, continuous with respect to the pair $ (y,\xi)$, and $ Q$-periodic as a function of the variable $ y$. Uniform continuity with respect to the $ x$ variable, as it is customary in the existing literature, is not required.

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