A note on some overdetermined elliptic problem

TL;DR

This paper introduces the concept of exceptional manifolds, explores their properties in two dimensions, provides examples, a construction method using complex analysis, and classifies such surfaces under certain conditions.

Contribution

It defines exceptional manifolds with harmonic functions satisfying boundary conditions, offers a construction algorithm, and classifies these surfaces in two dimensions.

Findings

Examples of exceptional manifolds are provided.

A construction algorithm for such surfaces is developed.

A classification theorem for these surfaces is proved.

Abstract

We define the notion of an exceptional manifold to be a flat Riemannian manifold with boundary which supports a positive harmonic function satisfying simultaneously a zero Dirichlet condition and a constant (nonzero) Neumann condtion at the boundary. We study the two-dimensional case: we present various examples and give a general construction algorithm of such surface by using complex analysis. We deduce a classification of all such surfaces assuming some further natural hypotheses and prove a Bernstein type theorem.