Extensions of the space of continuous functions and embedding theorems

A. K. Gushchin

Sbornik. Mathematics, Tome 211 (2020) no. 11, pp. 1551-1567

Résumé

The machinery of $s$-dimensionally continuous functions is developed for the purpose of applying it to the Dirichlet problem for elliptic equations. With this extension of the space of continuous functions, new generalized definitions of classical and generalized solutions of the Dirichlet problem are given. Relations of these spaces of $s$-dimensionally continuous functions to other known function spaces are studied. This has led to a new construction (seemingly more successful and closer to the classical one) of $s$-dimensionally continuous functions, using which new properties of such spaces have been identified. The embeddings of the space $C_{s,p}(\overline Q)$ in $C_{s',p'}(\overline Q)$ for $s'>s$ and $p'>p$, and, in particular, in $ L_q(Q)$ are proved. Previously, $W^1_2(Q)$ was shown to embed in $C_{n-1,2}(\overline Q)$, which secures the $(n-1)$-dimensional continuity of generalized solutions. In the present paper, the more general embedding of $W^1_r(Q)$ in $C_{s,p}(\overline Q)$ is verified and the corresponding exponents are shown to be sharp. Bibliography: 33 titles.