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Counterexamples to the well-posedness of  transmission boundary value problems for the Laplacian
Author(s):
Irina
Mitrea;
Katharine
Ott
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2037-2043.
MSC (2000):
Primary 45E05, 47A05;
Secondary 35J25, 42B20
Posted:
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]$](/proc/2007-135-07/S0002-9939-07-08686-8/gif-abstract0/img1.gif) for  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  spectra of such singular integrals.
References:
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- 1.
- R.R.Coifman, A.McIntosh and Y.Meyer, L'integrale de Cauchy définit un opérateur borné sur
 pour les courbes lipschitziennes, Annals of Math., 116 (1982), 361-387. MR 0672839 (84m:42027) - 2.
- L.Escauriaza and M.Mitrea, Transmission problems and spectral theory for singular integral operators on Lipschitz domains, J. Funct. Anal., 216 (2004), no. 1, 141-171. MR 2091359 (2005f:35065)
- 3.
- K.Hoffman, Banach spaces of analytic functions, Dover Publications, Inc., New York, 1988. MR 1102893 (92d:46066)
- 4.
- D.Mitrea, Layer potentials and Hodge decompositions in two dimensional Lipschitz domains, Math. Ann., 322 (2002), no. 1, 75-101. MR 1883390 (2003d:42024)
- 5.
- V.Y.Shelepov, On the index and spectrum of integral operators of potential type along Radon curves, Math. USSR Sbornik, 70, no. 1 (1991), 175-203. MR 1072296 (91i:47070)
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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:
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
Posted:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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