Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains

Anton, Ramona

Bulletin de la Société Mathématique de France, Tome 136 (2008) no. 1, pp. 27-65.

Nous considérons le problème de Cauchy pour l'équation de Schrödinger non linéaire sur un domaine du plan avec des conditions aux limites de Dirichlet. Nous prouvons que le problème est bien posé et qu'il existe une solution globale pour une non linéarité polynomiale défocalisante. La preuve repose sur une inégalité de Strichartz généralisée sur des variétés munies d'une métrique de Lipschitz.

We prove wellposedness of the Cauchy problem for the nonlinear Schrödinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.

Classification : 35Q55, 35Bxx, 81Q20
Keywords: nonlinear schrödinger, dispersive equations, Lipschitz metric
Mot clés : schrödinger non-linéaire, équations dispersives, métrique lipschitzienne

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