We study the corrector matrix to the conductivity equations. We show that if converges weakly to the identity, then for any laminate at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. We use this...
This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...
The theory of
compensated compactness of Murat and Tartar links the algebraic condition
of rank- convexity with the analytic condition of weak
lower
semicontinuity. The former is an algebraic
condition and therefore it is, in principle, very easy to use. However,
in applications of this theory, the need for an efficient classification of
rank- convex forms arises. In the present paper,
we define the concept of extremal -forms and characterize them
in the rotationally invariant jointly...
This paper is part of a larger project initiated with [2]. The
final aim of the present paper is to give bounds for the homogenized (or
effective) conductivity in two dimensional linear conductivity. The main focus is
therefore the periodic setting. We prove new variational principles that
are shown to be of interest in finding bounds on the homogenized
conductivity. Our results unify previous approaches by the second author and make
transparent the central role of quasiconformal mappings in all...
We study the corrector matrix to the conductivity equations. We show
that if converges weakly to the identity, then for any laminate
at almost every point. This simple property is shown to be false for
generic microgeometries if the dimension is greater than two in the work Briane [, to appear].
In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [
(2001) 155-171]. We use this property of laminates to prove that, in any...
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