Integral Operators in Sobolev Spaces on Domains with Boundary

✍ Scribed by Jörg Witte

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 460 KB
• Volume: 200
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19992000107

✦ Synopsis

Abstract

Properties of integral operators with weak singularities arc investigated. It is assumed that G ⊂ ℝ^n^ is a bounded domain. The boundary δ__G__ should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators \documentclass{article}\pagestyle{empty}\begin{document}$\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$\end{document} maps W~p~^k^(G) into W~p~^k+1^(G) and W~p~^k−1^(G) into W~p~^k/p^(G), respectively, and are bounded. Here θ ∈ S ⊂ ℝ^n^, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.