AVERAGING AND<i>G</i>-CONVERGENCE OF DIFFERENTIAL OPERATORS

Zhikov, V. V.; Козлов, С. М.; Oleinik, O A; Ngoan, Kha T'en

doi:10.1070/rm1979v034n05abeh003898

Cited by 267 publications

(122 citation statements)

References 21 publications

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“…By a composite made from materials and a2 we mean a mixture having fine scale structure, with perfect bonding at all material interfaces. To give a mathematical definition one can use the theory of random composites (see, e.g., [18,34]), the theory of //-convergence (also known as G-convergence; see, e.g., [12,32,36,37,43]), or the spatially periodic theory (see, e.g., [8,35]). However, the last point of view is the easiest to work with and is sufficient for proving bounds (for a rigorous proof of this point, see [18] in the random case and [10] in the general case of //-convergence).…”

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confidence: 99%

Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

Allaire¹,

Kohn²

1993

Quart. Appl. Math.

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Abstract. This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the "Hashin-Shtrikman variational principle" and the other on the "translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are "well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation. 0. Introduction. The macroscopic properties of a linearly elastic composite material are described by its tensor of effective moduli (Hooke's law) a*. This fourthorder tensor depends on the microgeometry of the mixture as well as on the elastic properties of the components. There is a large body of literature concerning the estimation of a* in terms of statistical information on the microstructure; see, e.g., [9,39,41].Recently a related but somewhat different question has received much attention: given a fixed collection of component materials, can one describe all composites o* achievable by mixing these components in prescribed volume fraction? Known as the "(7-closure problem", this question arises naturally from problems of structural optimization; see, e.g., [24,26,33]. A complete answer is available only in a few special cases; see, e.g., [13,25,27]. Much more is known about the analogous question

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Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

Allaire¹,

Kohn²

1993

Quart. Appl. Math.

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“…They lie on the theory of G-convergence, about which we can refer the reader to [ZKOT79] and [ZKO81]. We first assert a result concerning elliptic problems.…”

Section: Convergence Resultsmentioning

confidence: 99%

“…In this situation the theory of G-convergence can help us the get homogenized coefficients that will be piecewise constant (see [ZKOT79], see also [BLP78] on homogenization).…”

Section: The Choice Of the Piecewise Constant Coefficientsmentioning

confidence: 99%

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

Étoré

Lejay

2007

ESAIM: PS

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To cite this version:Pierre Etoré, Antoine Lejay. A Donsker theorem to simulate one-dimensional processes with measurable coefficients. ESAIM: Probability and Statistics, EDP Sciences, 2007, 11, pp.301-326 In this paper, we prove a Donker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.

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“…We finally explain in Section 7.2 how is our work related to works focused on the long time asymptotics of the heat kernel (see [KS00], [DZ00], [ZKON79]) and finally in Section 7.3 we state how Theorem 1 transposes to all graded nilmanifolds (subject which should be widely extended in a forthcoming article).…”

Section: Introduction and Claimsmentioning

confidence: 99%

The macroscopic sound of tori

Vernicos¹

2004

Pacific J. Math.

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Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes to infinity, and characterise the flat tori using the tools of homogenisation our conclusion being that "Macroscopically, one can hear the shape of a flat torus". We also show how in the two dimensional case we can recover earlier results by D. Burago, S. Ivanov and I. Babenko on the asymptotic volume.

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AVERAGING ANDG-CONVERGENCE OF DIFFERENTIAL OPERATORS

Cited by 267 publications

References 21 publications

Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

The macroscopic sound of tori

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