The stability for the Cauchy problem for elliptic equations

TL;DR

This paper establishes optimal stability results for the ill-posed Cauchy problem in elliptic equations, using the three-spheres inequality as a fundamental tool, with broad applicability under minimal assumptions.

Contribution

It introduces a unified approach to derive stability estimates for elliptic Cauchy problems using the three-spheres inequality, applicable in general settings.

Findings

Derived essentially optimal stability estimates

Unified framework based on three-spheres inequality

Applicable under minimal assumptions

Abstract

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.