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Anisotropic Sobolev Spaces and a Quasidistance Function

Anisotropic Sobolev Spaces and a Quasidistance Function ANISOTROPIC SOBOLEV SPACES AND A QUASIDISTANCE FUNCTION W. D. EVANS AND J. RAKOSNIK 1. Introduction Let D be a proper, non-empty, open subset of R^, N > 1. For k e N and p e [1, oo), it is well known that if an element / of the Sobolev space W*' (Q) is such that fd- EU{Q), (1.1) N v P where d(x) = dist (x, R - Q), then/ e Wl> {£l) the closure of q°(Q ) in H^ (Q) ; see, for example, [2, Theorem V.3.4]. This result was extended in [3] and [7] to weighted Sobolev spaces and to the spaces F' and B* in [8]. In [1] D. E. Edmunds and v g p q iP R. M. Edmunds considered the problem for the anisotropic Sobolev spaces Wj , N p kp where k = (k ,...,k )eN . They proved that if>e(l , oo), then/ e W^ if/ e W (Q) 1 N and /^-*-GL (Q) , where k = max {k ;i= \,2,...,N}. (1.2) m t r k P For the case p — 1 they required Q to be bounded and also that {D]f) d ~ i e L (Q) for all http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Anisotropic Sobolev Spaces and a Quasidistance Function

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/23.1.59
Publisher site
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Abstract

ANISOTROPIC SOBOLEV SPACES AND A QUASIDISTANCE FUNCTION W. D. EVANS AND J. RAKOSNIK 1. Introduction Let D be a proper, non-empty, open subset of R^, N > 1. For k e N and p e [1, oo), it is well known that if an element / of the Sobolev space W*' (Q) is such that fd- EU{Q), (1.1) N v P where d(x) = dist (x, R - Q), then/ e Wl> {£l) the closure of q°(Q ) in H^ (Q) ; see, for example, [2, Theorem V.3.4]. This result was extended in [3] and [7] to weighted Sobolev spaces and to the spaces F' and B* in [8]. In [1] D. E. Edmunds and v g p q iP R. M. Edmunds considered the problem for the anisotropic Sobolev spaces Wj , N p kp where k = (k ,...,k )eN . They proved that if>e(l , oo), then/ e W^ if/ e W (Q) 1 N and /^-*-GL (Q) , where k = max {k ;i= \,2,...,N}. (1.2) m t r k P For the case p — 1 they required Q to be bounded and also that {D]f) d ~ i e L (Q) for all

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1991

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