Embedding theorems and boundary-value problems for cusp domains

V. Gol'dshtein, M. J. Vasiltchik

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We study the Robin boundary-value problem for bounded domains with isolated singularities. Because trace spaces of space (D)on boundaries of such domains are weighted Sobolev spaces L2ξ (∂D), existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators I1 : W12) → L 2(D)and I2 : W12 (D) → L 2'ξ (∂D) i.e. on types of singularities. We obtain an exact description of weights ξ 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 I1,I2, we prove also that these Robin boundary-value problems with the spectral parameter are of Fredholm type.

Original languageEnglish
Pages (from-to)1963-1979
Number of pages17
JournalTransactions of the American Mathematical Society
Issue number4
StatePublished - 1 Apr 2010


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