A note on some overdetermined elliptic problem [article]

Frédéric Hélein, Frank Pacard
2010 arXiv   pre-print
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.
arXiv:1001.1060v1 fatcat:jxb54o4vvfbtvhgz6d4jofzwhu