Two-Dimensional Div-Curl Results: Application to the Lack of Nonlocal Effects in Homogenization

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.     

Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general hypothesis, we study the homogenization of non-linear non-monotone degenerate elliptic operators. We obtain some general homogenization result, which result is applied to the resolution of several concrete homogenization problems such as the periodic homogenization and the almost periodic homogenization problems. Our main tool is the theory of homogenization structures.     

Homogénéisation et compacité par compensation

In a recent paper S. Fabre, J. Mossino and the author obtained a homogenization result for a class of nonlinear problems with coefficients having specific dependence upon coordinates. Their method of proof used either Aubin's lemma or compensated compactness. In this Note, our aim is to prove that the above mentioned result can be extended to a larger class of problems, where the dependence upon coordinates is replaced by a weaker condition, in terms of divergence and curl.     

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

We establish quantitative homogenization, large‐scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well‐approximated by harmonic functions on ℝ^d. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher‐order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville‐type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|^k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil‐Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ. © 2018 Wiley Periodicals, Inc.     

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.     

Γ-convergence and homogenization of functionals in Sobolev spaces with variable exponents

This paper is devoted to homogenization and minimization problems for variational functionals in the framework of Sobolev spaces with continuous variable exponents. We assume that the sequence of exponents converges in the uniform metric and that the Lagrangian has a periodic microstructure. Then under natural coerciveness assumptions we prove a Γ-convergence result and, as a consequence, the convergence of minimizers (solutions to the corresponding Euler equations).     

Deterministic homogenization of integral functionals with convex integrands

In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence.     

Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients

The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.     

A note on the existence of extension operators for Sobolev spaces on periodic domains

In this note, we prove the existence of a family of extension operators for Sobolev spaces defined on ε-periodic domains. The norms of the operators are shown to be independent of ε. This extension theorem is relevant in the theory of homogenization for PDE's under flux boundary conditions.     

Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.     

An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains

The aim of this paper is to prove the existence of extension operators for SBV functions from periodically perforated domains. This result will be the fundamental tool to prove the compactness in a noncoercive homogenization problem.     

Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain Ω ^ε in ℝ ^N with isolated holes. While the space variable in the coefficients a ₀ and a is scaled with size ε (ε>0 a small parameter), the system of holes is scaled with ε ² size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.     

A link between sets of tensors stable under lamination and quasiconvexity

A link is found between quasiconvexity and the conditions for a set L of conductivity or elasticity tensors to be stable under lamination. These conditions, derived in the companion paper, are shown here to be equivalent to the condition that for every point B on the boundary of the set L an operator T_B dependent on the tangent plane and curvature of the set at B is a quasiconvex translation operator. A separate class of quasiconvex translation operators is obtained which are candidates for proving that L is stable under homogenization. The region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set L and has curvature at that point not greater than the curvature of L. The conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around L remains unresolved. It is proved that L can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their Legendre transforms. © 1994 John Wiley & Sons, Inc.     

Homogenization and corrector for the wave equation with discontinuous coefficients in time

In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded.     

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ ^N with isolated holes of size ɛ² (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.     

Homogenization of Fučík Eigencurves

In this work we study the convergence of an homogenization problem for half-eigenvalues and Fu?ík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.     

Comparison of hyperelastic 2‐D and 3‐D model foams at large deformations

The influence of the modeling dimension on the determination of effective properties for hyperelastic foams is investigated by means of regular 2‐D and 3‐D model foams. For calculating the effective stress‐strain relationships of both microstructures, a strain energy based homogenization procedure is employed. The results from numerical analyses show that with a 2‐D model foam the basic deformation mechanisms of the 3‐D model can be captured. Nevertheless, due to the distinct quantitative deviations found from the homogenization analyses, 3‐D modeling approaches should be used if quantitative predictions for the effective material properties are required. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Individual homogenization of nonlinear parabolic operators

In this article, we prove an individual homogenization result for a class of almost periodic nonlinear parabolic operators. The spatial and temporal heterogeneities are almost periodic functions in the sense of Besicovitch. The latter allows discontinuities and is suitable for many applications. First, we derive stability and comparison estimates for sequences of G-convergent nonlinear parabolic operators. Furthermore, using these estimates, the individual homogenization result is shown.     

Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales

Multiscale homogenization of nonlinear non-monotone degenerated parabolic operators is investigated. Under a periodicity assumption on the coefficients of the operators under consideration, we obtain by means of multiscale convergence method, an accurate homogenization result. It is also shown that in spite of the presence of several time scales the global homogenized problem is not a reiterated one.