$L^p$ Carleman estimates for elliptic boundary value problems and applications to the quantification of unique continuation

Dehman, Belhassen; Ervedoza, Sylvain; Thabouti, Lotfi

doi:10.5802/ahl.226

L^{p}

Carleman estimates for elliptic boundary value problems and applications to the quantification of unique continuation
[Estimées de Carleman

L^{p}

pour des problèmes aux limites elliptiques et applications à la quantification du prolongement unique]

Dehman, Belhassen ¹ ; Ervedoza, Sylvain ² ; Thabouti, Lotfi ^3,⁴

¹Université 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)
²Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence (France)
³Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence(France)
⁴Département de mathématiques, Faculté des sciences de Tunis, Université de Tunis El Manar, 2092 El Manar & Ecole Nationale d’Ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis (Tunisia)

Annales Henri Lebesgue, Tome 7 (2024), pp. 1603-1668

Abstract (VO)
Résumé

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 $Δ u = V u + W_{1} \cdot \nabla u + \div (W_{2} u)$ in terms of the norms of $V$ in $L^{q_{0}} (Ω)$ , of $W_{1}$ in $L^{q_{1}} (Ω)$ and of $W_{2}$ in $L^{q_{2}} (Ω)$ for $q_{0} \in (\frac{d}{2}, \infty]$ and $q_{1}$ and $q_{2}$ satisfying either $q_{1}, q_{2} > \frac{3 d - 2}{2}$ and $\frac{1}{q_{1}} + \frac{1}{q_{2}} < 4 (1 - \frac{1}{d}) / (3 d - 2)$ , or $q_{1}, q_{2} > \frac{3 d}{2}$ .

L’objectif de ce travail est de démontrer des estimations de Carleman $L^{p}$ globales pour l’opérateur Laplacien en dimension $d \geq 3$ . Notre stratégie repose sur des estimations de Carleman sur des bandes puis un recollement approprié des estimations locales et au bord obtenues grâce à un changement de variables. L’essentiel du travail consiste à prouver des estimations de Carleman dans la bande avec une fonction poids linéaire pour un opérateur du second ordre à coefficients dépendant linéairement de la variable normale. Cela est réalisé par la construction d’une paramétrice explicite pour l’opérateur conjugué, qui est estimée grâce à l’utilisation des théorèmes de restriction de Stein–Tomas. En application, nous déduisons des versions quantifiées de la propriété de prolongement unique pour les solutions de $Δ u = V u + W_{1} \cdot \nabla u + \div (W_{2} u)$ en termes des normes de $V$ dans $L^{q_{0}} (Ω)$ , de $W_{1}$ dans $L^{q_{1}} (Ω)$ et de $W_{2}$ dans $L^{q_{2}} (Ω)$ pour $q_{0} \in (\frac{d}{2}, \infty]$ et $q_{1}$ et $q_{2}$ satisfaisant soit $q_{1}, q_{2} > \frac{3 d - 2}{2}$ et $\frac{1}{q_{1}} + \frac{1}{q_{2}} < 4 (1 - \frac{1}{d}) / (3 d - 2)$ , soit $q_{1}, q_{2} > \frac{3 d}{2}$ .

Reçu le : 2023-07-02
Révisé le : 2024-07-22
Accepté le : 2024-07-22
Publié le : 2025-03-11

DOI : 10.5802/ahl.226

Classification : 35BXX, 35J25, 35B60, 35R05
Keywords: Carleman estimates, boundary value problem, elliptic equations, Fourier restriction theorems

Affiliations des auteurs :

Dehman, Belhassen ¹ ; Ervedoza, Sylvain ² ; Thabouti, Lotfi ^{3
,

4}

¹ 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)
² Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence (France)
³ Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, CNRS, Bordeaux INP, F-33400 Talence(France)
⁴ Département de mathématiques, Faculté des sciences de Tunis, Université de Tunis El Manar, 2092 El Manar & Ecole Nationale d’Ingénieurs de Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis (Tunisia)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {$L^p$ {Carleman} estimates for elliptic boundary value problems and applications to the quantification of unique continuation},
     journal = {Annales Henri Lebesgue},
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Dehman, Belhassen; Ervedoza, Sylvain; Thabouti, Lotfi. $L^p$ Carleman estimates for elliptic boundary value problems and applications to the quantification of unique continuation. Annales Henri Lebesgue, Tome 7 (2024), pp. 1603-1668. doi: 10.5802/ahl.226

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