no. 1
Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications
[Estimée de Strichartz globales pour l’équation de Schrödinger avec conditions au bord non triviales et applications]
Audiard, Corentin1
1 Sorbonne Université, Université Paris-Diderot SPC CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris (France)
Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 31-80.

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 H s , 0≤s≤2 et des données au bord dans un espace naturel ℋ s , il améliore les estimées de Strichartz locales en temps obtenues récemment par d’autres auteurs dans le cas des conditions de Dirichlet. Pour s≥1/2, la définition des conditions de compatibilité requiert une étude précise des espaces ℋ s . 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 ℋ s .

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 H s ,0≤s≤2 and boundary data in a natural space ℋ s . For s≥1/2, the issue of compatibility conditions requires a thorough analysis of the ℋ s 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 ℋ s space.

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DOI : 10.5802/aif.3238
Classification : 35Q41, 35G31, 35B45, 35B65
Keywords: Schrödinger equation, dispersive estimates, boundary conditions, Kreiss–Lopatinskii, compatibility condition
Mot clés : Équation de Schrödinger, estimation dispersives, conditions au bord, Kreiss–Lopatinskii, condition de compatibilité

Audiard, Corentin 1

1 Sorbonne Université, Université Paris-Diderot SPC CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Global {Strichartz} estimates for the {Schr\"odinger} equation with non zero boundary conditions and applications},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Audiard, Corentin. Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications. Annales de l'Institut Fourier, Tome 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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