Stability in an overdetermined problem for the Green's function

TL;DR

This paper investigates the stability of reconstructing a planar domain from the normal derivative of its Green's function, providing polynomial stability estimates using conformal mapping theory.

Contribution

It introduces new stability estimates for domain reconstruction from Green's function derivatives, leveraging conformal mappings for the first time in this context.

Findings

Derived polynomial stability estimates in Holder norms

Established a link between Green's function derivatives and domain shape

Applied conformal mapping theory to inverse problems

Abstract

In the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green's function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings, we derive stability estimates of polynomial type in Holder norms.