Estimations de Carleman $L^p$ pour des problèmes au bord elliptiques et applications à la quantification du prolongement unique

DEHMAN, Belhassen; ERVEDOZA, Sylvain; THABOUTI, Lotfi

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DEHMAN, Belhassen
Université de Tunis El Manar, Faculté des Sciences de Tunis, 2092 El Manar & Ecole Nationale d'Ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia.

ERVEDOZA, Sylvain
Institut de Mathématiques de Bordeaux [IMB]

THABOUTI, Lotfi
Institut de Mathématiques de Bordeaux [IMB]

Langue

Document de travail - Pré-publication

Ce document a été publié dans

2023-06-28

Résumé en anglais

The aim of this work is to prove global $L^p$ Carleman estimates for the Laplace operator in dimension $d \geq 3$. Our strategy relies on precise Carleman estimates in strips, and a suitable gluing of local and boundary estimates obtained through a change of variables. The delicate point and most of the work thus consists in proving Carleman estimates in the strip with a linear weight function for a second order operator with coefficients depending linearly on the normal variable. This is done by constructing an explicit parametrix for the conjugated operator, which is estimated through the use of Stein Tomas restriction theorems. As an application, we deduce quantified versions of the unique continuation property for solutions of $\Delta u = V u + W_1 \cdot \nabla u + \div(W_2 u)$ in terms of the norms of $V$ in $L^{q_0}(\Omega)$, of $W_1$ in $L^{q_1}(\Omega)$ and of $W_2$ in $L^{q_2}(\Omega)$ for $q_0 \in (d/2, \infty]$ and $q_1$ and $q_2$ satisfying either $q_1, \, q_2 > (3d-2)/2$ and $1/q_1 + 1/q_2< 4 (1-1/d)/(3d-2)$, or $q_1, \, q_2 > 3d/2$.< Réduire

Mots clés en anglais

Carleman estimates

boundary value problem

elliptic equations

Fourier restriction theorems

Project ANR

Nouvelles directions en contrôle et stabilisation: Contraintes et termes non-locaux - ANR-20-CE40-0009

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251