Doubling Inequality and Strong Unique Continuation for an Elliptic Transmission Problem

TL;DR

This paper proves the strong unique continuation property for elliptic equations with discontinuous coefficients across an interface, using a doubling inequality from a Carleman estimate, aiding inverse problems involving inclusions.

Contribution

It establishes SUCP at interfaces with jump discontinuities for elliptic equations, advancing understanding of inverse problems with measurable inclusions.

Findings

Proved SUCP at interfaces with jump discontinuities.

Derived a doubling inequality from a Carleman estimate.

Facilitated future inverse problem solutions involving inclusions.

Abstract

We investigate the Strong Unique Continuation Property (SUCP) for elliptic equations with piecewise Lipschitz coefficients exhibiting jump discontinuities across a regular interface. We prove SUCP at the interface using a doubling inequality derived from a Carleman estimate with a singular weight. This result is intended as a first step toward solving the inverse problem of estimating the size of an unknown, merely measurable, inclusion inside a conductor from boundary measurements.