On the Compactness of Embeddings of Weighted Sobolev Spaces on a Domain with Irregular Boundary

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).