A straightforward proof of Carleman estimate for second order elliptic operator and a three sphere inequality

TL;DR

This paper presents a simple proof of a Carleman estimate for second order elliptic operators with Lipschitz coefficients and uses it to establish a three sphere inequality for solutions.

Contribution

The paper introduces a straightforward proof technique for Carleman estimates applicable to elliptic operators with Lipschitz coefficients, facilitating the derivation of three sphere inequalities.

Findings

Established a simple proof of Carleman estimate for elliptic operators

Derived a three sphere inequality for solutions to elliptic equations

Applicable to operators with Lipschitz continuous coefficients

Abstract

In this paper we provide a simple proof of a Carleman estimate for a second order elliptic operator $P$ with Lipschitz leading coefficients. We apply such a Carleman estimate to derive a three sphere inequality for solutions to equation $Pu=0$