Lower bounds of Dirichlet eigenvalues for degenerate elliptic operators and degenerate Schrödinger operators

Hua Chen, Peng Luo, and Shuying Tian

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Submission date: 19. Aug. 2013
Pages: 17
Bibtex
MSC-Numbers: 35P05, 35P20
Keywords and phrases: Lower bounds for Dirichlet eigenvalues, finite degenerate elliptic operators, infinite degenerate elliptic operators, singular potential term
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Abstract:
Let X = (X₁,X₂,,X_m) be a system of real smooth vector fields defined in an open domain ⊂ ℝⁿ, Ω ⊂⊂ be a bounded open subset in ℝⁿ with smooth boundary ∂Ω, △_X = ∑ _j=1^mX_j². In this paper, if λ_j is the j^th Dirichlet eigenvalue for the degenerate elliptic operator -△_X (or the degenerate Schrödinger operator -△_X + V ) on Ω, we deduce respectively that the lower estimates for the sums ∑ _j=1^kλ_j in both cases for the operator -△_X to be finitely degenerate (i.e. the Hörmander condition is satisfied) or infinitely degenerate (i.e. the Hörmander condition is not satisfied).