Symmetries in an overdetermined problem for the Green's function

TL;DR

This paper investigates how to reconstruct planar domains from Green's function data, establishing existence, uniqueness, and symmetry conditions using conformal mapping theory, and relates boundary curvature to Green's function derivatives.

Contribution

It introduces new symmetry results and a formula linking boundary curvature to Green's function derivatives in the context of domain reconstruction.

Findings

Existence and uniqueness of domain reconstruction from Green's function data.

Non-spherical symmetry results for reconstructed domains.

A formula relating boundary curvature to the normal derivative of Green's function.

Abstract

We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness, (non-spherical) symmetry results, and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function.