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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Strichartz estimates for Dirichlet-wave equations in two dimensions with applications


Authors: Hart F. Smith, Christopher D. Sogge and Chengbo Wang
Journal: Trans. Amer. Math. Soc. 364 (2012), 3329-3347
MSC (2010): Primary 35L71; Secondary 35B45, 35L20
Published electronically: January 31, 2012
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Abstract: We establish the Strauss conjecture for nontrapping obstacles
when the spatial dimension $ n$ is two. As pointed out by Hidano, Metcalfe, Smith, Sogge, and Zhou (2010) this case is more subtle than $ n=3$ or $ 4$ due to the fact that the arguments in the papers of the first two authors (2000), Burq (2000) and Metcalfe (2004), showing that local Strichartz estimates for obstacles imply global ones, require that the Sobolev index, $ \gamma $, equals $ 1/2$ when $ n=2$. We overcome this difficulty by interpolating between energy estimates ($ \gamma =0$) and ones for $ \gamma =\frac 12$ that are generalizations of Minkowski space estimates of Fang and the third author (2006), (2011), the second author (2008) and Sterbenz (2005).


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Additional Information

Hart F. Smith
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195

Christopher D. Sogge
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05607-8
PII: S 0002-9947(2012)05607-8
Keywords: Strichartz estimates, Strauss conjecture, obstacles
Received by editor(s): December 14, 2010
Received by editor(s) in revised form: April 17, 2011
Published electronically: January 31, 2012
Additional Notes: The authors were supported in part by the NSF. The third author was supported in part by NSFC 10871175 and 10911120383.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.