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Distance Functions and Orlicz-Sobolev Spaces

Published online by Cambridge University Press:  20 November 2018

D. E. Edmunds  and
R. M. Edmunds
D. E. Edmunds
Affiliation:
University of Sussex, Brighton, U.K
R. M. Edmunds
Affiliation:
University College, Cardiff, U.K
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Let ∧ be a bounded, non-empty, open subset of Rn and given any x in Rn, let

let k ∊ N and suppose that p ∞ (1, ∞). It is known (c.f. e.g. [4]) that if u belongs to the Sobolev space WKp(∧) and u/dk ∊ Lp(∧), then . Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

Under appropriate smoothness conditions on the boundary of ∧, m and d are equivalent, and thus may in this case be characterized as the subspace of W1,2(∧) consisting of all functions u ∊ W1,2(∧) such that u/d ∊ L2(∧). Further results in this direction are given in [5] and [9]. Moreover, if m is the mean distance function in the sense of [2], then it turns out that

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986 , pp. 1181 - 1198
Copyright
Copyright © Canadian Mathematical Society 1986

References

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