Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

TL;DR

This paper establishes three-ball inequalities and propagation of smallness for complex second order elliptic equations with discontinuous Lipschitz coefficients, aiding in size estimation problems using boundary Cauchy data.

Contribution

It introduces new three-ball inequalities and propagation of smallness results for elliptic equations with discontinuous complex coefficients, extending previous Carleman estimate techniques.

Findings

Derived three-ball inequalities for complex elliptic equations

Proved propagation of smallness in the presence of discontinuities

Applied estimates to size estimation problems with boundary data

Abstract

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [FVW].