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Title of Journal: Calc Var

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Abbravation: Calculus of Variations and Partial Differential Equations

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Springer Berlin Heidelberg

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DOI

10.1007/0-306-46924-3_2

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1432-0835

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Lower bounds of Dirichlet eigenvalues for some deg

Authors: Hua Chen Peng Luo
Publish Date: 2015/07/03
Volume: 54, Issue: 3, Pages: 2831-2852
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Abstract

Let Omega be a bounded open domain in mathbb Rn with smooth boundary and X=X 1 X 2 ldots X m be a system of real smooth vector fields defined on Omega with the boundary partial Omega which is noncharacteristic for X If X satisfies the Hörmander’s condition then the vector fields is finite degenerate and the sum of square operator triangle X=sum j=1mX j2 is a finitely degenerate elliptic operator otherwise the operator triangle X is called infinitely degenerate If lambda j is the jth Dirichlet eigenvalue for triangle X on Omega then this paper shall study the lower bound estimates for lambda j Firstly by using the subelliptic estimate directly we shall give a simple lower bound estimates of lambda j for general finitely degenerate triangle X which is polynomial increasing in j Secondly if triangle X is socalled Grushin type degenerate elliptic operator then we can give a precise lower bound estimates for lambda j Finally by using logarithmic regularity estimate for infinitely degenerate elliptic operator triangle X we prove that the lower bound estimates of lambda j will be logarithmic increasing in j

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