Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

We present a general numerical method for calculating effective elastic properties of periodic structures based on the homogenization method. Some concrete numerical examples are presented.

We reformulate and extend G. Nguetseng’s notion of two-scale convergence by means of a variable transformation, and outline some of its properties. We approximate two-scale derivatives, and extend this convergence to spaces of differentiable functions. The two-scale limit of derivatives of bounded sequences in the Sobolev spaces $W^{1,p}\left({\mathbb{R}}^N\right)$, $L_{rot}^2{\left({\mathbb{R}}^3\right)}^3$, $L_{div}^2{\left({\mathbb{R}}^3\right)}^3$ and $W^{2,p}\left({\mathbb{R}}^N\right)$ is then characterized. The two-scale limit behaviour of the potentials of a two-scale convergent sequence of irrotational fields is finally studied.

The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization...

The main goal is to show that the pointwise Osserman four-dimensional pseudo-Riemannian manifolds (Lorentzian and manifolds of neutral signature $(--++)$) can be characterized as self dual (or anti-self dual) Einstein manifolds. Also, examples of pointwise Osserman manifolds which are not Osserman are discussed.

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). ...

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ${\partial }_{t}u+{div}_{y}A(y,u)-{\Delta }_{y}u=0$, where $y\in {ℝ}^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^1$ norm like a self-similar profile for large times. The proof uses a time and space change of variables which is...

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form $$-div\left(a\left(T_1\left(\frac{x}{{\epsilon }_1}\right){\omega }_1,T_2\left(\frac{x}{{\epsilon }_2}\right){\omega }_2,\nabla u_{\epsilon }^{\omega }\right)\right)={\lambda }_{\epsilon }^{\omega }𝒞\left(u_{\epsilon }^{\omega }\right).$$ It is shown, under certain structure assumptions on the random map $a({\omega }_1,{\omega }_2,\xi )$, that the sequence $\{{\lambda }_{\epsilon }^{\omega ,k},u_{\epsilon }^{\omega ,k}\}$ of $k$th eigenpairs converges to the $k$th eigenpair $\{{\lambda }^k,u^k\}$ of the homogenized eigenvalue problem $$-\mathrm{div}\left(b\left(\nabla u\right)\right)=\lambda \overline{𝒞}\left(u\right).$$ For the case of $p$-Laplacian type maps we characterize $b$ explicitly.

The author is partially supported by: M. U. R. S. T. Prog. Nazionale “Problemi e Metodi nella Teoria delle Equazioni Iperboliche”.We treat the oscillatory problem for semilinear wave equation. The oscillatory initial data are of the type u(0, x) = h(x) + ε^(σ+1) * e^(il(x)/ε) * b0 (ε, x) ∂t u(0, x) = ε^σ * e^(il(x)/ε) * b1(ε, x). By using suitable variants of Strichartz estimate we extend the results from [6] on a priori estimates of the approximations of geometric optics.The main improvement...

Dans cet article nous présentons quelques problèmes et résultats d’homogénéisation non locale pour certaines équations de type dégénéré. Nous considérons des équations de transport, une équation des ondes dégénérée et une équation différentielle de Riccati, et nous décrivons dans chacun des cas les effets non locaux induits par homogénéisation. Nous donnons aussi quelques résultats sur l’analyse mathématique des équations des fluides miscibles en milieu poreux.