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

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

Belhassen Dehman (1) , Sylvain Ervedoza (2) , Lotfi Thabouti (2)

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 IMB - Institut de Mathématiques de Bordeaux

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 $q_1, \, q_2 > 3d/2$.

Mots clés: Carleman estimates; boundary value problem; elliptic equations; Fourier restriction theorems

Domaines: Mathématiques [math]

Belhassen Dehman, Sylvain Ervedoza, Lotfi Thabouti. $L^p$ Carleman estimates for elliptic boundary value problems and applications to the quantification of unique continuation. 2023. ⟨hal-04148255⟩