08-204 Rafael de la Llave, Enrico Valdinoci

Symmetry for a Dirichlet-Neumann problem arising in water waves (267K, pdf uuencoded) Oct 27, 08

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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.

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