Extensions and Imbeddings

✍ Scribed by P Koskela

• Book ID: 102590532
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 257 KB
• Volume: 159
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3331

✦ Synopsis

We establish a connection between the Sobolev imbedding theorem and the extendability of Sobolev functions. As applications we give geometric criteria for extendability and give a result on the dependence of the extension property on the exponent p.

1998 Academic Press

Theorem A. Suppose that W 1, p (D) / Ä C 0, 1&n p (D ) for a fixed p>n. Then there is a bounded extension operator of W 1, q (D) into W 1, q (R n ) for all q>p.

Thus the Sobolev imbedding theorem essentially implies that Sobolev functions can be extended to all of R n . For p n, there can be no such result. Indeed, for example W 1, p (D) imbeds into L p V (D), p V =2 pÂ(2& p), for 1 p<2, and one has the Trudinger type of inequality corresponding to p=2 provided D is the unit disk minus a radius. For this domain D, the functions in W 1, p (D) cannot be extended to functions in W 1, q (R 2 ) for article no.