%0 Journal Article
%T A note on some overdetermined elliptic problem
%A Fr¨¦d¨¦ric H¨¦lein
%A Laurent Hauswirth
%A Frank Pacard
%J Mathematics
%D 2010
%I arXiv
%X 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.
%U http://arxiv.org/abs/1001.1060v1