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

Robert Lipton; Tadele Mengesha

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 5, page 1121-1146
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Lipton, Robert, and Mengesha, Tadele. "Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.5 (2012): 1121-1146. <http://eudml.org/doc/273265>.

@article{Lipton2012,
abstract = {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.},
author = {Lipton, Robert, Mengesha, Tadele},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {L∞norms; nonlinear composition; weak limits; material design; homogenization; -convergence; piecewise constant coefficient matrix; local corrector; periodic homogenization; laminate composite; graded microstructure; optimal design},
language = {eng},
number = {5},
pages = {1121-1146},
publisher = {EDP-Sciences},
title = {Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization},
url = {http://eudml.org/doc/273265},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Lipton, Robert
AU - Mengesha, Tadele
TI - Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 5
SP - 1121
EP - 1146
AB - 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.
LA - eng
KW - L∞norms; nonlinear composition; weak limits; material design; homogenization; -convergence; piecewise constant coefficient matrix; local corrector; periodic homogenization; laminate composite; graded microstructure; optimal design
UR - http://eudml.org/doc/273265
ER -

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