Rafael de la Llave, Enrico Valdinoci
Symmetry for a Dirichlet-Neumann problem
arising in water waves
(267K, pdf uuencoded)

ABSTRACT.  Given a smooth $u:\R^n\rightarrow\R$, say $u=u(y)$,
we consider $\overline u=\overline u(x,y)$
to be a solution of
$$ \left\{
\begin{matrix}
\Delta \overline u =0 & {\mbox{ for any $(x,y)\in
(0,1)\times\R^n$,}}\\
\overline u(0,y)= u(y) & 
{\mbox{ for any $y\in
\R^n$,}}
\\
\overline u_x (1,y)=0&
{\mbox{ for any $y\in
\R^n$.}}
\end{matrix}
\right. $$
We define the Dirichlet-Neumann operator $
({\mathcal{L}} u)(y)=\overline u_x (0,y)$
and we prove
a symmetry result for equations of the form
$({\mathcal{L}} u)(y)=f(u(y))$.
In particular,
bounded, monotone
solutions
in $\R^2$ are proven to
depend only on one Euclidean variable.