$L^p$ 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

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, Volume 7 (2024), pp. 1603-1668.

Metadata

DOI10.5802/ahl.226

Keywords Carleman estimates, boundary value problem, elliptic equations, Fourier restriction theorems

Abstract

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 $q_{1}, q_{2} > \frac{3 d}{2}$ .

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