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1.
Marc Briane 《Archive for Rational Mechanics and Analysis》2006,182(2):255-267
The paper deals with the asymptotic behaviour as ε → 0 of a two-dimensional conduction problem whose matrix-valued conductivity a
ε
is ε-periodic and not uniformly bounded with respect to ε. We prove that only under the assumptions of equi-coerciveness and L
1-boundedness of the sequence a
ε
, the limit problem is a conduction problem of same nature. This new result points out a fundamental difference between the
two-dimensional conductivity and the three-dimensional one. Indeed, under the same assumptions of periodicity, equi-coerciveness
and L
1-boundedness, it is known that the high-conductivity regions can induce nonlocal effects in three (or greater) dimensions. 相似文献
2.
The notion of a Hall matrix associated with a possibly anisotropic conducting material in the presence of a small magnetic
field is introduced. Then, for any material having a microstructure we prove a general homogenization result satisfied by
the Hall matrix in the framework of the H-convergence of Murat–Tartar. Extending a result of Bergman, we show that the Hall matrix can be computed from the corrector
associated with the homogenization problem when no magnetic field is present. Finally, we give an example of a microstructure
for which the Hall matrix is positive isotropic almost everywhere, while the homogenized Hall matrix is negative isotropic. 相似文献
3.
Marc Briane Juan Casado–Díaz 《Calculus of Variations and Partial Differential Equations》2008,33(4):463-492
In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type
, where A
n
is a symmetric positive definite matrix-valued function and μ
n
is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A
n
we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ
n
and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of [11,23] in which A
n
is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional
diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear.
However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates,
the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an
explicit formula for the limit energy specifying the kernel of the nonlocal term. 相似文献
4.
5.
In this article we study the asymptotic behaviour as tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\epsilon$-periodic bounded open set of . The period cell of is equal to where is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer such that the n-th non-zero eigenvalue of the spectral problem in satisfies , the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical
assumptions on the perforated domain .
Received: 20 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001 相似文献
6.
In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L1 combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two. More precisely, in the two-dimensional case we extend the Murat–Tartar H-convergence which holds for uniformly bounded and equi-coercive conductivity sequences, as well as the compactness result which holds for bounded and equi-integrable conductivity sequences in L1. Our homogenization results are based on extensions of the classical div-curl lemma, which are also specific to the dimension two. 相似文献
7.
Homogenized laws for sequences of high-contrast two-phase non-symmetric conductivities perturbed by a parameter are derived in two and three dimensions. The parameter characterizes the antisymmetric part of the conductivity for an idealized model of a conductor in the presence of a magnetic field. In dimension two an extension of the Dykhne transformation to non-periodic high conductivities permits to prove that the homogenized conductivity depends on through some homogenized matrix-valued function obtained in the absence of a magnetic field. This result is improved in the periodic framework thanks to an alternative approach, and illustrated by a cross-like thin structure. Using other tools, a fiber-reinforced medium in dimension three provides a quite different homogenized conductivity. 相似文献
8.
9.
This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach, and the density of the continuous functions in the domain of the Γ-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three. 相似文献
10.
In this paper, we study the two-dimensional Hall effect in a highly heterogeneous conducting material in the low magnetic field limit. Extending Bergman's approach in the framework of H-convergence we obtain the effective Hall coefficient which only depends on the corrector of the material resistivity in the absence of a magnetic field. A positivity property satisfied by the effective Hall coefficient is then deduced from the homogenization process. An explicit formula for the effective Hall coefficient is derived for anisotropic interchangeable two-phase composites. 相似文献