Classification of the solutions to an overdetermined problem in the plane

TL;DR

This paper classifies solutions to a specific overdetermined elliptic problem in the plane with finite connectivity by linking them to minimal surfaces, providing a comprehensive understanding of their structure.

Contribution

It establishes a one-to-one correspondence between solutions of the elliptic problem and certain minimal surfaces, advancing the classification of such solutions.

Findings

Solutions correspond to a specific class of minimal surfaces

Complete classification achieved for finite connectivity cases

New link between elliptic problems and geometric surface theory

Abstract

We classify the solutions to an overdetermined elliptic problem in the plane in the finite connectivity case. This is achieved by establishing a one-to-one correspondence between the solutions to this problem and a certain type of minimal surfaces.