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ORLICZ-SOBOLEV SPACES WITH ZERO BOUNDARY VALUES ON METRIC SPACES

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Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html

ISBN 1083-0464

Issue 1 July 2004, pp. 10 – 32

Submitted: September 10, 2003. Published: July 1, 2004

ORLICZ-SOBOLEV SPACES WITH ZERO BOUNDARY VALUES ON METRIC SPACES

NOUREDDINE A¨ISSAOUI

Abstract. In this paper we study two approaches for the defi- nition of the first order Orlicz-Sobolev spaces with zero boundary values on arbitrary metric spaces. The first generalization, de- noted byMΦ1,0(E), where E is a subset of the metric space X, is defined by the mean of the notion of the trace and is a Banach space when the N-function satisfies the ∆2condition. We give also some properties of these spaces. The second, following another def- inition of Orlicz-Sobolev spaces on metric spaces, leads us to three definitions that coincide for a large class of metric spaces and N- functions. These spaces are Banach spaces for any N-function.

A.M.S. (MOS) Subject Classification Codes.46E35, 31B15, 28A80.

Key Words and Phrases. Orlicz spaces, Orlicz-Sobolev spaces, modulus of a family of paths, capacities.

Ecole Normale Sup´erieure, B.P 5206 Ben Souda, F`es, Maroc [email protected]

Copyright c2004 by Cameron University 1

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