On Strichartz estimates for Schrödinger operators in compact manifolds with boundary

Authors:
Matthew D. Blair, Hart F. Smith and Christopher D. Sogge

Journal:
Proc. Amer. Math. Soc. **136** (2008), 247-256

MSC (2000):
Primary 35Q40, 35B65; Secondary 35Q55, 35A17

Published electronically:
October 12, 2007

MathSciNet review:
2350410

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove local Strichartz estimates with a loss of derivatives over compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

**1.**Anton, R.*Strichartz inequalities for Lipschitz metrics on manifolds and the nonlinear Schrödinger equation on domains*. Preprint.**2.**J. Bourgain,*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations*, Geom. Funct. Anal.**3**(1993), no. 2, 107–156. MR**1209299**, 10.1007/BF01896020**3.**N. Burq, P. Gérard, and N. Tzvetkov,*Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds*, Amer. J. Math.**126**(2004), no. 3, 569–605. MR**2058384****4.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****5.**J. Ginibre and G. Velo,*On the global Cauchy problem for some nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré Anal. Non Linéaire**1**(1984), no. 4, 309–323 (English, with French summary). MR**778977****6.**Markus Keel and Terence Tao,*Endpoint Strichartz estimates*, Amer. J. Math.**120**(1998), no. 5, 955–980. MR**1646048****7.**Herbert Koch and Daniel Tataru,*Dispersive estimates for principally normal pseudodifferential operators*, Comm. Pure Appl. Math.**58**(2005), no. 2, 217–284. MR**2094851**, 10.1002/cpa.20067**8.**Hart F. Smith,*Spectral cluster estimates for 𝐶^{1,1} metrics*, Amer. J. Math.**128**(2006), no. 5, 1069–1103. MR**2262171****9.**Hart F. Smith and Christopher D. Sogge,*On the 𝐿^{𝑝} norm of spectral clusters for compact manifolds with boundary*, Acta Math.**198**(2007), no. 1, 107–153. MR**2316270**, 10.1007/s11511-007-0014-z**10.**Gigliola Staffilani and Daniel Tataru,*Strichartz estimates for a Schrödinger operator with nonsmooth coefficients*, Comm. Partial Differential Equations**27**(2002), no. 7-8, 1337–1372. MR**1924470**, 10.1081/PDE-120005841**11.**Robert S. Strichartz,*Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations*, Duke Math. J.**44**(1977), no. 3, 705–714. MR**0512086****12.**Daniel Tataru,*Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation*, Amer. J. Math.**122**(2000), no. 2, 349–376. MR**1749052****13.**Daniel Tataru,*Phase space transforms and microlocal analysis*, Phase space analysis of partial differential equations. Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, pp. 505–524. MR**2208883****14.**Michael E. Taylor,*Pseudodifferential operators and nonlinear PDE*, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. MR**1121019**

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Additional Information

**Matthew D. Blair**

Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Email:
mblair@math.jhu.edu

**Hart F. Smith**

Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195

Email:
hart@math.washington.edu

**Christopher D. Sogge**

Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Email:
sogge@jhu.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-09114-9

Received by editor(s):
October 31, 2006

Published electronically:
October 12, 2007

Additional Notes:
The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.