Memory Effects and $\boldsymbol{\Gamma}$-Convergence: A Time Dependent Case

Abstract: We study the asymptotic behavior of the energy functional associated to the parametric equation $\partial_t u_\varepsilon(t, x)$ $+ a_\varepsilon(t, x) g (u_\varepsilon(t, x)) =f(t, x)$, $u_\varepsilon(0, x) =u_0(x)$. Techniques of Young measures are improved in order to characterize the $\Gamma$-limit of the sequence of energy functionals in terms of the oscillations of $a_\varepsilon$. We assume some sort of independence between time and the oscillating character of $a_\varepsilon$. The example of a periodic mixture of two materials with coefficients analytic in time is presented.

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