Geodesic
On the Compactness of Embeddings of Weighted Sobolev Spaces on a~Domain with Irregular Boundary
Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 72-93

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Sufficient conditions are established for the compactness of the embedding of the weighted Sobolev spaces $W_p^s$, $s\in\mathbb N$, into the weighted Lebesgue space $L_q$ for domains with irregular boundaries, in particular, for a cusp domain. The conditions imposed on the domain are formulated in simple geometrical terms (of a degenerate flexible cone). 
@article{TRSPY_2001_232_a8,
     author = {O. V. Besov},
     title = {On the {Compactness} of {Embeddings} of {Weighted} {Sobolev} {Spaces} on {a~Domain} with {Irregular} {Boundary}},
     journal = {Informatics and Automation},
     pages = {72--93},
     publisher = {mathdoc},
     volume = {232},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a8/}
}
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O. V. Besov. On the Compactness of Embeddings of Weighted Sobolev Spaces on a~Domain with Irregular Boundary. Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 72-93. http://geodesic.mathdoc.fr/item/TRSPY_2001_232_a8/