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Mat. Sb., 2020, Volume 211, Number 11, Pages 54–71 (Mi msb9415)  

Extensions of the space of continuous functions and embedding theorems

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: 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.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).


DOI: https://doi.org/10.4213/sm9415

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English version:
Sbornik: Mathematics, 2020, 211:11, 1551–1567

Bibliographic databases:

UDC: 917.956.223+517.982.272
MSC: Primary 46E15, 46E35; Secondary 35J60
Received: 23.03.2020

Citation: A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Mat. Sb., 211:11 (2020), 54–71; Sb. Math., 211:11 (2020), 1551–1567

Citation in format AMSBIB
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