An Overdetermined Problem in Potential Theory

TL;DR

This paper characterizes specific planar domains admitting harmonic functions satisfying both Dirichlet and Neumann boundary conditions, revealing only three such domains under certain assumptions, and explores higher-dimensional analogs.

Contribution

It provides a classification of domains with a special harmonic function, identifying only three solutions in the plane and analyzing higher-dimensional cases.

Findings

Only three domains admit the roof function in the plane.

The nontrivial example in four dimensions lacks an axially symmetric analog.

Higher-dimensional analogs do not contain their own axis of symmetry.

Abstract

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.