Symmetry for a Dirichlet–Neumann problem arising in water waves
Rafael de la Llave, Enrico Valdinoci · Mathematical Research Letters · 2009
Given a smooth u : R n → R, say u = u(y), we consider u = u(x, y) to be a solution of 8 < :for any y ∈ R n , ux(1, y) = 0 for any y ∈ R n .We define the Dirichlet-Neumann operator (L u)(y) = ux(0, y) and we prove a symmetry result for equations of the form (L u)(y) = f (u(y)).In particular, bounded, monotone solutions in R 2 are proven to depend only on one Euclidean variable.