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.
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.
Keywords: nonlinear schrödinger, dispersive equations, Lipschitz metric
Mot clés : schrödinger non-linéaire, équations dispersives, métrique lipschitzienne
@article{BSMF_2008__136_1_27_0, author = {Anton, Ramona}, title = {Strichartz inequalities for {Lipschitz} metrics on manifolds and nonlinear {Schr\"odinger} equation on domains}, journal = {Bulletin de la Soci\'et\'e Math\'ematique de France}, pages = {27--65}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {136}, number = {1}, year = {2008}, doi = {10.24033/bsmf.2548}, mrnumber = {2415335}, zbl = {1157.35100}, language = {en}, url = {http://archive.numdam.org/articles/10.24033/bsmf.2548/} }
TY - JOUR AU - Anton, Ramona TI - Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains JO - Bulletin de la Société Mathématique de France PY - 2008 SP - 27 EP - 65 VL - 136 IS - 1 PB - Société mathématique de France UR - http://archive.numdam.org/articles/10.24033/bsmf.2548/ DO - 10.24033/bsmf.2548 LA - en ID - BSMF_2008__136_1_27_0 ER -
%0 Journal Article %A Anton, Ramona %T Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains %J Bulletin de la Société Mathématique de France %D 2008 %P 27-65 %V 136 %N 1 %I Société mathématique de France %U http://archive.numdam.org/articles/10.24033/bsmf.2548/ %R 10.24033/bsmf.2548 %G en %F BSMF_2008__136_1_27_0
Anton, Ramona. Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains. Bulletin de la Société Mathématique de France, Volume 136 (2008) no. 1, pp. 27-65. doi : 10.24033/bsmf.2548. http://archive.numdam.org/articles/10.24033/bsmf.2548/
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