no. 1
Global Strichartz estimates for the Schrödinger equation with non zero boundary conditions and applications
Audiard, Corentin1
1Sorbonne Université, Université Paris-Diderot SPC CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris (France)
Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 31-80

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.

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 .

Received:
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Accepted:
Published online:
DOI: 10.5802/aif.3238
Classification: 35Q41, 35G31, 35B45, 35B65
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

1 Sorbonne Université, Université Paris-Diderot SPC CNRS, Laboratoire Jacques-Louis Lions, LJLL, F-75005 Paris (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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
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