Weighted Function Spaces on Domains and Degenerate Elliptic Differential Equations

Abstract

In [I, 3.2.3.] we introduced the weighted function spaces $B_{p,q}^8$(Ω,ϱ ^μ (x), ϱ ^ν(x)) and $H_{p}^8$ Ω,ϱ ^μ (x), ϱ ^ν(x)), where Ω is an arbitrary domain on R _n and ϱ(x) is a weight function which degen˜rates near the boundary ∂ Ω of Ω and at infinity (if Ω is unbounded). Furthermore 1 <p<∞,1≦q≦∞ and8≦O. We treated these spaces in some detail and used them to study degenerate elliptic differential equations in [1, Chapter 6]. In H. Triebel [11] we extended these considerations to the spaces $B_{p,p}^8$(Ω,ϱ ^μ (x), ϱ ^ν(x)) with O <p<∞.It is the aim of this chapter to give an idea of these spaces and their applications to elliptic differential equations. For that purpose, we restrict ollrselves to model cases, both for the spaces (in particular for the domain Ω) and the elliptic differential operators. Essentially, we follow the summaries given in [S, ,R. 4 and R. 6]. A reader who is interested in proofs and generalizations is asked to consult [I] and H. Triebel [11].