Overdetermined elliptic problems in topological disks

TL;DR

This paper presents a novel method using the Poincare-Hopf index theorem to classify solutions of overdetermined elliptic problems in disk-shaped domains, extending classical uniqueness results to nonlinear PDEs.

Contribution

The paper introduces a new classification approach for overdetermined elliptic problems in topological disks, generalizing Hopf's theorem to nonlinear elliptic PDEs.

Findings

Method based on Poincare-Hopf index theorem for solution classification

Application to well-known nonlinear elliptic PDEs

Extension of Hopf's uniqueness theorem to nonlinear context

Abstract

We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.