L 1 Properties of Two Classes of Singular Second Order Elliptic Operators
Mingqi Pang · Journal of the London Mathematical Society · 1988
Let Ω be a region in ℝN, let σ be a positive C∞-function on Ω and let H be the elliptic operator in L2 (Ω, σλ dx) defined formally by Hu = −σ−λ div {σμ grad u), λ,μ∈ℝ. We investigate how the spectral properties of H depend on λ and μ for the following cases : (i) Ω is a bounded C∞-domain in ℝN and σ(x) ∼ dist (x, ∂Ω), (ii) Ω = ℝN and σ(x) ∼ (1 + |x|). We show that certain L1 and L2 spectral properties of H depend only on λ and μ and not on the particular σ chosen. Our results are summarized in Tables 1 and 2 at the end of the paper.