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

Robert Lipton; Tadele Mengesha

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

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

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topLipton, Robert, and Mengesha, Tadele. "Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1121-1146. <http://eudml.org/doc/222152>.

@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},

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},

month = {2},

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/222152},

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

DA - 2012/2//

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/222152

ER -

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