Given a smooth u : ℝn → ℝ, say u = u(y), we consider ū = ū(x, y) to be a solution of △ū = 0 for any (x, y) ∈ (0, 1) × ℝn, ū(0, y) = u(y) for any y ∈ ℝn, ūx(1, y) = 0 for any y ∈ ℝn. We define the Dirichlet-Neumann operator (Lu)(y) = ūx(0, y) and we prove a symmetry result for equations of the form (Lu)(y) = f(u(y)). In particular, bounded, monotone solutions in ℝ2 are proven to depend only on one Euclidean variable.
Symmetry for a Dirichlet-Neumann problem arising in water waves / R. de la Lave, E. Valdinoci. - In: MATHEMATICAL RESEARCH LETTERS. - ISSN 1073-2780. - 16:5(2009), pp. 909-918.
Symmetry for a Dirichlet-Neumann problem arising in water waves
E. ValdinociPrimo
2009
Abstract
Given a smooth u : ℝn → ℝ, say u = u(y), we consider ū = ū(x, y) to be a solution of △ū = 0 for any (x, y) ∈ (0, 1) × ℝn, ū(0, y) = u(y) for any y ∈ ℝn, ūx(1, y) = 0 for any y ∈ ℝn. We define the Dirichlet-Neumann operator (Lu)(y) = ūx(0, y) and we prove a symmetry result for equations of the form (Lu)(y) = f(u(y)). In particular, bounded, monotone solutions in ℝ2 are proven to depend only on one Euclidean variable.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
