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Characterisation of zero trace functions in variable exponent Sobolev spaces
Author(s) -
Edmunds D. E.,
Nekvinda Aleš
Publication year - 2017
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201600102
Subject(s) - mathematics , sobolev space , bounded function , exponent , zero (linguistics) , trace (psycholinguistics) , constant (computer programming) , space (punctuation) , variable (mathematics) , pure mathematics , type (biology) , function (biology) , mathematical analysis , ecology , philosophy , linguistics , evolutionary biology , computer science , biology , programming language
It is well known that if u belongs to the Sobolev spaceW 1 , p( Ω ) , where Ω is an open subset of R N and p ∈ ( 1 , ∞ ) , then u ∈ W 0 1 , p( Ω )if u / d belongs to weakL p ( Ω ) , where d ( x ) = dist x , ∂ Ω . Results of this type are given here for Sobolev spaces with a variable exponent p , under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log‐Hölder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that u ∈ W 0 1 , p( Ω )if and only if u ∈ W 1 , p( Ω )and u / d ∈ L 1 ( Ω ) .