Quantitative unique continuation for non-regular perturbations of the Laplacian

TL;DR

This paper provides explicit quantitative estimates for the unique continuation property of solutions to elliptic equations with non-regular potentials, refining Carleman estimates and extending previous methods to broader potential classes.

Contribution

It introduces refined Carleman estimates and combines them with Wolff's argument to quantify unique continuation for elliptic equations with less regular potentials.

Findings

Explicit bounds on solutions in terms of potential norms

Refined Carleman estimates for non-regular potentials

Extension of unique continuation results to broader potential classes

Abstract

In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$, where $d \geq 3$. Here, $V \in L^{q_0}$, $W_1 \in L^{q_1}$, and $W_2 \in L^{q_2}$ with $q_0 > d/2$, $q_1 > d$, and $q_2 > d$. Our aim is to provide an explicit quantification of the unique continuation property with respect to the norms of the potentials. To achieve this, we revisit the Carleman estimates established in [Dehman-Ervedoza-Thabouti-2023] and prove a refined version of them, and we combine them with an argument due to T. Wolff introduced in [Wolff-1992] for the proof of unique continuation for solutions of equations of the form $\Delta u = V u + W_1 \cdot \nabla u$.