On the non-homogeneous boundary value problem for Schrödinger equations

Corentin Audiard

doi:10.3934/dcds.2013.33.3861

Article Contents

2013, Volume 33, Issue 9: 3861-3884. Doi: 10.3934/dcds.2013.33.3861

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On the non-homogeneous boundary value problem for Schrödinger equations

Corentin Audiard^1,

1.
UPMC Univ Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received: June 30, 2012

Revised: January 31, 2013

Published: August 2013

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Abstract

We study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When $\Omega$ is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If $\Omega^c$ is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01, 35A02, 35B30, 35B45.

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