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.

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.

We consider the Neumann Laplacian with constant magnetic field on a regular domain in ${ℝ}^2$. Let $B$ be the strength of the magnetic field and let ${\lambda }_1\left(B\right)$ be the first eigenvalue of this Laplacian. It is proved that $B\mapsto {\lambda }_1\left(B\right)$ is monotone increasing for large $B$. Together with previous results of the authors, this implies the coincidence of all the “third” critical fields for strongly type 2 superconductors.

Using an approximation method, we show the existence of solutions for some noncooperative elliptic systems defined on an unbounded domain.

The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L^2(a,b) and the infinitesimal generator of a C_0-semigroup of bounded linear operators.

This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis.

Pour un système parabolique de lois de conservation, nous considérons le problème mixte, dans le domaine $x\>0$. Pour une condition de Dirichlet, le système admet en général des solutions stationnaires $U\left(x\right)$, qui tendent vers une limite en $+\infty$. Ce sont les profils des couches limites, dans l’approximation du second ordre, pour le système hyperbolique du premier ordre sous-jacent. La stabilité de cette couche limite est liée à la stabilité linéaire asymptotique de $U$. On étudie celle-ci au moyen d’une fonction d’Evans,...