Traces of Sobolev functions with one square integrable directional derivative

✍ Scribed by M. Gregoratti

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 181 KB
• Volume: 29
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.669

✦ Synopsis

Abstract

We consider the Sobolev spaces of square integrable functions v, from ℝ^n^ or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑∂~𝓁~v. In these spaces we define closed trace operators on the boundaries ∂Q and on the hyperplanes {r~𝓁~ = z}, z ∈ ℝ{0}, which turn out to be possibly unbounded with respect to the usual L^2^‐norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L^2^, we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ℝ^n^ and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.