A quantitative Carleman estimate for second order elliptic operators

TL;DR

This paper establishes a quantitative Carleman estimate for second order elliptic operators with Lipschitz coefficients, explicitly detailing dependence on key parameters, which enhances understanding of unique continuation properties.

Contribution

It provides a novel, explicit quantitative Carleman estimate for elliptic operators with Lipschitz coefficients, including uniform bounds on the weight function.

Findings

Explicit dependence on Lipschitz and ellipticity constants

Uniform bounds on the weight function

Applicability to complex-valued functions in W^{2,2}

Abstract

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of arbitrary radius. The novelty of this Carleman estimate is that we establish an explicit dependence on the Lipschitz and ellipticity constants, the dimension of the space and the radius of the ball. In particular we provide a uniform and quantitative bound on the weight function for a class of elliptic operators given explicitly in terms of ellipticity and Lipschitz constant.