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Representation formulas for L∞norms of weakly convergent sequences of gradient fields in homogenization

Published online by Cambridge University Press:  13 February 2012

Robert Lipton  and
Tadele Mengesha
Robert Lipton
Affiliation:
Department of Mathematics, Louisiana State University, 384 Lockett Hall, Baton Rouge, LA 70803-4918, USA. lipton@math.lsu.edu
Tadele Mengesha
Affiliation:
Department of Mathematics, Louisiana State University, 384 Lockett Hall, Baton Rouge, LA 70803-4918, USA. lipton@math.lsu.edu
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Abstract

We examine the composition of the L∞ norm with weaklyconvergent sequences of gradient fields associated with the homogenization of second orderdivergence form partial differential equations with measurable coefficients. Here thesequences of coefficients are chosen to model heterogeneous media and are piecewiseconstant and highly oscillatory. We identify local representation formulas that in thefine phase limit provide upper bounds on the limit superior of theL∞ norms of gradient fields. The local representationformulas are expressed in terms of the weak limit of the gradient fields and localcorrector problems. The upper bounds may diverge according to the presence of roughinterfaces. We also consider the fine phase limits for layered microstructures and forsufficiently smooth periodic microstructures. For these cases we are able to provideexplicit local formulas for the limit of the L∞ norms of theassociated sequence of gradient fields. Local representation formulas for lower bounds areobtained for fields corresponding to continuously graded periodic microstructures as wellas for general sequences of oscillatory coefficients. The representation formulas areapplied to problems of optimal material design.

Keywords

L∞normsnonlinear compositionweak limitsmaterial designhomogenization
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1121 - 1146
Copyright
© EDP Sciences, SMAI, 2012

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