Symmetry for a Dirichlet–Neumann problem arising in water waves

Given a smooth $u:\mathbb{R}^n\rightarrow\mathbb{R}$, say $u=u(y)$, we consider $\overline u=\overline u(x,y)$ to be a solution of$$ \left\{\begin{matrix}\Delta \overline u =0 & {\mbox{ for any $(x,y)\in(0,1)\times\mathbb{R}^n$,}}\\\overline u(0,y)= u(y) &{\mbox{ for any $y\in\mathbb{R}^n$,}}\\\overline u_x (1,y)=0&{\mbox{ for any $y\in\mathbb{R}^n$.}}\end{matrix}\right. $$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 $\mathbb{R}^2$ are proven to depend only on one Euclidean variable.

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Published 1 January 2009