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Measures of non–compactness of classical embeddings of Sobolev spaces

✍ Scribed by Stanislav Hencl

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
221 KB
Volume
258
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let Ω be an open subset of ℝ^n^ and let p ∈ [1, n). We prove that the measure of non–compactness of the Sobolev embedding W^k,p^~0~(Ω) → L^p*^(Ω) is equal to its norm. This means that the entropy numbers of this embedding are constant and equal to the norm. The same is true, when λ~n~(Ω) is small enough, for the embedding of W^1,n^~0~(Ω) into the Orlicz space with Young function exp(t^n/(n−1^) − 1. The position is different for the embedding of W^1,p^~0~(J) in C^0,1−1/p^($ \bar J $), J equals; (0, 1), when p ∈ (1,∞): in this case the measure of non–compactness is less than the norm. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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