On the correctness of boundary value problems in the mechanics of continuous media

The paper considers the properties of vector analogs of the Sobolev spaces, $\mathbf W_p^1$, which appear in the study of various models of continuous media. Korn's inequality, proved in the paper, makes it possible to reduce the problem of compactness of the imbedding operators in these spaces to the scalar case and, consequently, to apply the scalar imbedding theorem of S. L. Sobolev. Heretofore, Korn's inequality in the general form was known only for $p=2$.
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