✦ LIBER ✦

Topology and Sobolev Spaces

✍ Scribed by Haim Brezis; Yanyan Li

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 296 KB
Volume: 183
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.2000.3736

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✦ Synopsis

Consider the Sobolev class W 1, p (M, N) where M and N are compact manifolds. We present some sufficient conditions which guarantee that W 1, p (M, N) is pathconnected. We also discuss cases where W 1, p (M, N) admits more than one component. There are still a number of open problems, especially concerning the values of p where a change in homotopy classes occurs.

2001 Academic Press

0. Introduction

Let M and N be compact 1 connected oriented smooth Riemannian manifolds with or without boundary. Throughout the paper we assume that dim M 2 but dim N could possibly be one, for example N=S 1 is of interest. Our functional framework is the Sobolev space W 1, p (M, N) which is defined by considering N as smoothly embedded in some Euclidean space R K and then

with 1 p< . W 1, p (M, N) is equipped with the standard metric d(u, v)=&u&v& W 1, p . Our main concern is to determine whether or not

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