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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Counterexamples to the well-posedness of $ L^p$ transmission boundary value problems for the Laplacian


Authors: Irina Mitrea and Katharine Ott
Journal: Proc. Amer. Math. Soc. 135 (2007), 2037-2043
MSC (2000): Primary 45E05, 47A05; Secondary 35J25, 42B20
Published electronically: February 28, 2007
MathSciNet review: 2299477
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Abstract: In this note we show that the well-posedness range $ p\in(1,2]$ for $ L^p$ transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp. Our approach relies on Mellin transform techniques for singular integrals naturally associated with the transmission problems and on a careful analysis of the $ L^p$ spectra of such singular integrals.


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Additional Information

Irina Mitrea
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: im3p@virginia.edu

Katharine Ott
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: kott@virginia.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08686-8
PII: S 0002-9939(07)08686-8
Keywords: Transmission boundary value problem, sector, spectrum
Received by editor(s): January 22, 2006
Published electronically: February 28, 2007
Additional Notes: The first author was supported in part by NSF Grant DMS 0547944 and a University of Virginia FEST Grant.
The second author was supported in part by an Aerospace Graduate Research Fellowship
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.