Necessary and sufficient conditions of compactness of certain embeddings of Sobolev spaces

V. I. Burenkov^ab, T. V. Tararykova<sup>cb

a</sup> V.A. Steklov Institute of Mathematics, Russian Academy of Sciences, 42 Vavilov St, 117966 Moscow, Russia
^b S.M. Nikol'skii Institute of Mathematics, RUDN University, 6 Miklukho Maklay St, 117198 Moscow, Russia
^c Cardiff School of Mathematics, Cardiff University, Senghennydd Road, CF24 4AG Cardiff, United Kingdom

Abstract: Necessary and sufficient conditions on an open set $\Omega\subset \mathbb{R}^n$ are obtained ensuring that for $l,m\in\mathbb{N}_0$, $m < l$ the embedding $\mathring{W}_\infty^l(\Omega)\subset W_\infty^m(\Omega)$ is compact, where $W_\infty^m(\Omega)$ is the Sobolev space and $\mathring{W}_\infty^l(\Omega)$ is the closure in $W_\infty^l(\Omega)$ of the space of all infinitely continuously differentiable functions on $\Omega$ with supports compact in $\Omega$.

Keywords and phrases: Sobolev spaces, pre-compact sets, embeddings of Sobolev spaces.

MSC: 46B25, 46B50, 47B38

Received: 01.09.2019

Language: English

DOI: 10.32523/2077-9879-2019-10-4-14-23

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