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Remarques sur l’observabilité pour l’équation de Laplace

Kim-Dang Phung (2003)

ESAIM: Control, Optimisation and Calculus of Variations

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....

Remarques sur l'observabilité pour l'équation de Laplace

Kim-Dang Phung (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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.

Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

Robert Lipton, Tadele Mengesha (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L∞ norms of gradient fields. The...

Representation formulas for L∞ norms of weakly convergent sequences of gradient fields in homogenization

Robert Lipton, Tadele Mengesha (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit...

Resolution of the time dependent Pn equations by a Godunov type scheme having the diffusion limit

Patricia Cargo, Gérald Samba (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model...

Résonances de Rayleigh en dimension 2

Didier Gamblin (2004)

Bulletin de la Société Mathématique de France

Nous étudions les résonances de Rayleigh créées par un obstacle strictement convexe à bord analytique en dimension 2. Nous montrons qu’il existe exactement deux suites de résonances ( z k , + ) et ( z k , - ) convergeant exponentiellement vite vers l’axe réel dans un voisinage polynomial de l’axe réel, et exponentiellement proches d’une suite de quasimodes réels. De plus, k - 1 ℜ z k , ± est un symbole analytique d’ordre 0 en la variable k - 1 dont on donne le premier terme du développement. Nous construisons pour cela des quasimodes...

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