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Transactions of the American Mathematical Society

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Embedding theorems and boundary-value problems for cusp domains

Authors: V. Gol'dshtein and M. Ju. Vasiltchik
Journal: Trans. Amer. Math. Soc. 362 (2010), 1963-1979
MSC (2000): Primary 46E35, 35J25
Published electronically: November 13, 2009
MathSciNet review: 2574883
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Abstract: We study the Robin boundary-value problem for bounded domains with isolated singularities. Because trace spaces of space $ W_{2}^{1}(D)$ on boundaries of such domains are weighted Sobolev spaces $ L^{2,\xi}(\partial D)$, existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $ I_{1}:W_{2}^{1}(D)\rightarrow L^{2}(D)$ and $ I_{2}:W_{2}^{1}(D)\rightarrow L^{2,\xi}(\partial D)$ i.e. on types of singularities. We obtain an exact description of weights $ \xi$ for bounded domains with `outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators. Using compactness of embedding operators $ I_{1},I_{2}$, we prove also that these Robin boundary-value problems with the spectral parameter are of Fredholm type.

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Additional Information

V. Gol'dshtein
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P. O. Box 653, Beer Sheva, 84105, Israel

M. Ju. Vasiltchik
Affiliation: Department of Mathematics, Novosibirsk Technical University, Novosibirsk, Russia

Received by editor(s): October 9, 2007
Published electronically: November 13, 2009
Additional Notes: The first author was supported in part by the Israel Science Foundation grant
The second author was partially supported by the Russian Foundation for Basic Research (grant 06-01-00735)
Article copyright: © Copyright 2009 American Mathematical Society