We consider the Schrödinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time Strichartz estimates (for Dirichlet boundary conditions), we obtain global Strichartz estimates for initial data in and boundary data in a natural space . For , the issue of compatibility conditions requires a thorough analysis of the space. As an application we solve nonlinear Schrödinger equations and construct global asymptotically linear solutions for small data. A discussion is included on the appropriate notion of scattering in this framework, and the optimality of the space.
On considère l’équation de Schrödinger sur le demi espace en dimension arbitraire pour une classe de conditions au bord non homogènes, incluant les conditions de Dirichlet, Neumann, et « transparentes ». Le principal résultat consiste en des estimations de Strichartz globales pour des données initiales , et des données au bord dans un espace naturel , il améliore les estimées de Strichartz locales en temps obtenues récemment par d’autres auteurs dans le cas des conditions de Dirichlet. Pour , la définition des conditions de compatibilité requiert une étude précise des espaces . En application, on résout des équations de Schrödinger non linéaires, et on construit des solutions dispersives globales si les données sont petites. On discute également le sens précis donné à « solution dispersive », ainsi que la question de l’optimalité de l’espace .
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3238
Keywords: Schrödinger equation, dispersive estimates, boundary conditions, Kreiss–Lopatinskii, compatibility condition
Mots-clés : Équation de Schrödinger, estimation dispersives, conditions au bord, Kreiss–Lopatinskii, condition de compatibilité
Audiard, Corentin 1

@article{AIF_2019__69_1_31_0, author = {Audiard, Corentin}, title = {Global {Strichartz} estimates for the {Schr\"odinger} equation with non zero boundary conditions and applications}, journal = {Annales de l'Institut Fourier}, pages = {31--80}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {1}, year = {2019}, doi = {10.5802/aif.3238}, zbl = {07067399}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3238/} }
TY - JOUR AU - Audiard, Corentin TI - Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications JO - Annales de l'Institut Fourier PY - 2019 SP - 31 EP - 80 VL - 69 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3238/ DO - 10.5802/aif.3238 LA - en ID - AIF_2019__69_1_31_0 ER -
%0 Journal Article %A Audiard, Corentin %T Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications %J Annales de l'Institut Fourier %D 2019 %P 31-80 %V 69 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3238/ %R 10.5802/aif.3238 %G en %F AIF_2019__69_1_31_0
Audiard, Corentin. Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications. Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 31-80. doi : 10.5802/aif.3238. https://aif.centre-mersenne.org/articles/10.5802/aif.3238/
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