A note on some overdetermined elliptic problem
Mathematical Physics
2010-01-11 v1 Analysis of PDEs
Differential Geometry
math.MP
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.
Cite
@article{arxiv.1001.1060,
title = {A note on some overdetermined elliptic problem},
author = {Frédéric Hélein and Laurent Hauswirth and Frank Pacard},
journal= {arXiv preprint arXiv:1001.1060},
year = {2010}
}