Nous quantifions la propriété de continuation unique pour le laplacien dans un domaine borné quand la condition aux bords est a priori inconnue. Nous établissons une estimation de dépen-dance de type logarithmique suivant la terminologie de John [5]. Les outils utilisés reposent sur les inégalités de Carleman et les techniques des travaux de Robbiano [8, 11]. Aussi, nous déterminons en application de l’inégalité d’observabilité obtenue un coût du contrôle approché pour un problème elliptique modèle....

We consider the Laplace equation in a smooth bounded domain. We prove logarithmic estimates, in the sense of John [5] of solutions on a part of the boundary or of the domain without known boundary conditions. These results are established by employing Carleman estimates and techniques that we borrow from the works of Robbiano [8,11]. Also, we establish an estimate on the cost of an approximate control for an elliptic model equation.

Dans cet exposé, on décrit un travail effectué sous la direction de J. Sjöstrand. On prouve des majorations et des minorations du nombre de résonances d’un opérateur de Schrödinger semi-classique $P=-h^2\Delta +V\left(x\right)$ dans des petits disques centrés en $E_0\>0$, une valeur critique de $p(x,\xi )={\xi }^2+V\left(x\right)$.

We analyse the spectral phase diagram of Schrödinger operators $T+\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $iid$ random variables. The main result is a criterion for the emergence of absolutely continuous $\left(ac\right)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $ac$ spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of$\sim T$. Incorporating...

We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of $m$-charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset $X\subseteq {ℝ}^n$ satisfying a mild geometric condition, there is no uniformly continuous representation operator for $m$-charges in $X$.

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...


We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain...

Tetrahedral finite $C^0$-elements of the Hermite type satisfying the maximum angle condition are presented and the corresponding finite element interpolation theorems in the maximum norm are proved.

Let $\alpha >0$, $\lambda ={\left(2\alpha \right)}^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h\left(x\right)={\int }_{S^{n-1}}{e}^{\lambda \langle x,y\rangle }d\sigma \left(y\right)$. We characterize all subsets $M$ of ${ℝ}^n$ such that $$\underset{x\in {ℝ}^n}{inf}\frac{u\left(x\right)}{h\left(x\right)}=\underset{x\in M}{inf}\frac{u\left(x\right)}{h\left(x\right)}$$ for every positive solution $u$ of the Helmholtz equation on ${ℝ}^n$. A closely related problem of representing functions of $L_1\left(S^{n-1}\right)$ as sums of blocks of the form $e^{\lambda \langle x_k,.\rangle }/h\left(x_k\right)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

Material and shape derivatives for solutions to the Dirichlet Laplacian in a half-space are derived by an application of the speed method. The proposed method is general and can be used for shape sensitivity analysis in unbounded domains for the Neumann Laplacian as well as for the elasticity boundary value problems.

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is sharp.