Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Random Dirichlet problem: Scalar darcy's law

Using Ergodic Theory and Epiconvergence notion, we study the rate of convergence of solutions relative to random Dirichlet problems in domains ofR ^d with random holes whose size tends to 0. This stochastic analysis allows to extend the results already obtained in the corresponding periodic case.     

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F _n , , defined in L ²(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F _n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F _n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.     

Diffusive stability for periodic metric graphs

We consider a nonlinear diffusion equation on an infinite periodic metric graph. We prove that the terms which are irrelevant w.r.t. linear diffusion on the real line are irrelevant w.r.t. linear diffusion on the periodic metric graph, too. The proof is based on L¹‐ estimates combined with Bloch wave analysis for periodic metric graphs.     

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. Under convexity and p  -growth conditions (p>1$p>1$), it is proved that all such possible effective energy densities obtained by a Γ-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.     

Concentration phenomena for neutronic multigroup diffusion in random environments

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton–Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.     

Homogenization in general periodically perforated domains by a spectral approach

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     

Reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media

We present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients, which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures. Moreover, we study the Γ-convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence-free fields applying the properties of multiscale Young measures associated with sequences of divergence-free fields.     

Polarization tensors and effective properties of anisotropic composite materials

In this paper, we present a systematic scheme for derivations of asymptotic expansions including higher-order terms, with estimates, of the effective electrical conductivity of periodic dilute composites in terms of the volume fraction occupied by the inclusions. The conductivities of the inclusion and the matrix may be anisotropic. Our derivations are based on layer potential techniques, and valid for high contrast mixtures and inclusions with Lipschitz boundaries. The asymptotic expansion is given in terms of the polarization tensor and the volume fraction of the inclusions. Important properties, such as symmetry and positivity, of the anisotropic polarization tensors are derived.     

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c^?$c^{?}$ such that for each wave speed c≤c^?$c\le c^{?}$, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c^?$c\le c^{?}$ are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c^?$c>c^{?}$.     

On exchange axioms for valuated matroids and valuated delta-matroids

Two further equivalent axioms are given for valuations of a matroid. Let M = (V,B) be a matroid on a finite setV with the family of basesB. For :BR the following three conditions are equivalent: A similar result is obtained for valuations of a delta-matroid.This work was done while the author was at Forschungsinstitut für Diskrete Mathematik, Universität Bonn.     

Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects

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.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

Undecidability of automorphism groups

We give a simple (and easy to apply) technique that gives the undecidability of the theory of many automorphism groups: Let G be a group of automorphisms of a structure. Suppose that is not the identity and has no non-singleton finite orbits. If the centraliser of g is transitive on the support of g and satisfies a further technical condition, then the subgroup generated by g is equal to the double centraliser of g. Thus if G contains such an element g that is conjugate to all its positive powers, then one can interpret addition and multiplication of natural numbers in the theory of G using the parameter g; consequently, G has undecidable theory. Received: 9 October 2000 / in final form: 2 October 2001 / Published online: 29 April 2002     

Orientations of chain groups

We prove generalizations to chain groups, of Minty's Arc Colouring Lemma and its extension, the well-known Farkas Lemma. In these the orientation of the edges is replaced by an arbitrary chain.A function on a chain groupN isrepresentable if there exists a chainR such that (X)=R·X for allXN. Anorientation is a chain with values ±1. We prove that for a regular chain group a linear function that is representable by an orientation for each chainXN locally, is representable by an orientation globally.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients

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.     

Weighted estimates for stationary Navier-Stokes equations

We show Morrey-type estimates for the weak solution of the periodic Navier-Stokes equations in dimensionN, 5 <N < 10. ForN < 8, we prove the existence of a maximum solution.     

Two-scale convergence of some integral functionals

Nguetseng’s notion of two-scale convergence is reviewed, and some related properties of integral functionals are derived. The coupling of two-scale convergence with convexity and monotonicity is then investigated, and a two-scale version is provided for compactness by strict convexity. The div-curl lemma of Murat and Tartar is also extended to two-scale convergence, and applications are outlined.