Convergence Relative to a Microstructure : Properties, Optimal Bounds and Application

In this work, we study a new notion involving convergence of microstructures represented by matrices $B^\epsilon$ related to the classical $H$-convergence of $A^\epsilon$. It incorporates the interaction between the two microstructures. This work is about its effects on various aspects : existence, examples, optimal bounds on emerging macro quantities, application etc. Five among them are highlighted below : $(1)$ The usual arguments based on translated inequality, $H$-measures, Compensated Compactness etc for obtaining optimal bounds are not enough. Additional compactness properties are needed. $(2)$ Assuming two-phase microstructures, the bounds define naturally four optimal regions in the phase space of macro quantities. The classically known single region in the self-interacting case , namely $B^\epsilon= A^\epsilon$ can be recovered from them, a result that indicates we are dealing with a true extension of the $\mathcal{G}$-closure problem. $(3)$ Optimality of the bounds is not immediate because of (a priori) non-commutativity of macro-matrices, an issue not present in the self-interacting case. Somewhat surprisingly though, commutativity follows a posteriori. $(4)$ From the application to "Optimal Oscillation-Dissipation Problems", it emerges that oscillations and dissipation can co-exist optimally and the microstructures behind them need not be the same though they are closely linked. Furthermore, optimizers are found among $N$-rank laminates with interfaces. This is a new feature. $(5)$ Explicit computations in the case of canonical microstructures are performed, in which we make use of $H$-measure in a novel way.