Volume 46, Number 5, September-October 2012
|Page(s)||1121 - 1146|
|Published online||13 February 2012|
Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization
Department of Mathematics, Louisiana State
University, 384 Lockett
Hall, Baton Rouge,
Revised: 8 April 2011
We examine the composition of the L∞ 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∞ 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 microstructures. For these cases we are able to provide explicit local formulas for the limit of the L∞ 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.
Mathematics Subject Classification: 35J15 / 49N60
Key words: L∞norms / nonlinear composition / weak limits / material design / homogenization
© EDP Sciences, SMAI, 2012
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