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

Robert Lipton; Tadele Mengesha

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

Robert Lipton, Tadele Mengesha (2012)

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

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We examine the composition of the 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 ...

Which electric fields are realizable in conducting materials?

Marc Briane, Graeme W. Milton, Andrejs Treibergs (2014)

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

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In this paper we study the realizability of a given smooth periodic gradient field ∇ defined in R, in the sense of finding when one can obtain a matrix conductivity such that ∇ is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇ is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot...

The periodic unfolding method for a class of parabolic problems with imperfect interfaces

Zhanying Yang (2014)

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

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In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with -periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative a function of order with ≤ −1. We give the homogenization results which include those obtained by Jose in [54 (2009) 189–222]. We also get the corrector...

Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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We study integral functionals constrained to divergence-free vector fields in on a thin domain, under standard -growth and coercivity assumptions, 1 ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject...

-convergence and absolute minimizers for supremal functionals

Thierry Champion, Luigi De Pascale, Francesca Prinari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, we prove that the approximants naturally associated to a supremal functional -converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer...

Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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Hexahedral H(div) and H(curl) finite elements

Richard S. Falk, Paolo Gatto, Peter Monk (2011)

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

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We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a...

Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

Micol Amar, Virginia De Cicco, Nicola Fusco (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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New -lower semicontinuity and relaxation results for integral functionals defined in BV() are proved, under a very weak dependence of the integrand with respect to the spatial variable . More precisely, only the lower semicontinuity in the sense of the -capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to . Under this further...