Representation formulas for $L^\infty$ norms of weakly convergent sequences of gradient fields in homogenization

TL;DR

This paper derives local representation formulas for the $L^ Infty$ norms of gradient fields in homogenization, providing bounds and explicit formulas for microstructures, with applications to optimal material design.

Contribution

It introduces new local representation formulas for $L^ Infty$ norms of gradient sequences in homogenization, including explicit formulas for certain microstructures and bounds for rough interfaces.

Findings

Explicit formulas for layered and smooth periodic microstructures.

Upper bounds on $L^ Infty$ norms can diverge with rough interfaces.

Representation formulas aid in optimal material design.

Abstract

We examine the composition of the $L^{\infty}$ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the $L^{\infty}$ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the gradient fields and local corrector problems. The upper bounds may diverge according to the presence of rough interfaces. We also consider the fine phase limits for layered microstructures and for sufficiently smooth periodic microsturctures. For these cases we are able to provide explicit local formulas for the limit of the $L^\infty$ norms of the associated sequence of gradient fields. Local representation formulas for lower bounds are obtained for fields corresponding to continuously graded periodic microstructures as well as for general sequences of oscillatory coefficients. The representation formulas are applied to problems of optimal material design.