CHEN Hua,CHEN Hong-ge,LI Jin-ning

(School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

1 Introduction and Main Results

For n≥2,we consider the following Dirichlet eigenvalue problem

on a bounded open domainΩ⊂Rn,with smooth boundary∂Ω,whereis a degenerate elliptic operator generated by a system of real vector fields X=(X1,X2,···,Xn),namely,

We assume that the system of real vector fields X=(X1,X2,···,Xn)is defined in Rnby Xj=µj(x)∂xj,whereµ1,···,µnare real continuous nonnegative functions in Rnsatisfying following assumptions:

(H1)µ1=1,andµj(x)=µj(x1,···,xj−1)for j=2,···,n.

(H2)For each j=1,···,n,µj∈C1(RnΠ),where

(H3)µj(x)>0 and xk(∂xkµj)(x)≥0 for all x∈RnΠ,1≤k≤j−1,j=2,...,n.Furthermore,µj(x1,···,−xk,···,xj−1)=µj(x1,···,xk,···,xj−1)for all 1≤k≤j−1,j=2,···,n.

(H4)There exists a constantσj,k≥0 such thatholds for all 1≤k≤j−1,j=2,···,n.

Then,we define some positive constantsε1,...,εnas

and an index Q as

We remark that assumption(H1)allows us to write the operatorin the form=Recently,Kogoj and Lanconelli in[3]studied such degenerate elliptic operators under the following additional assumption:

(H6)There exists a group of dilations(δt)t>0,

with 1=α1≤ α2≤ ···≤ αn such thatµi isδt homogeneous of degreeαi−1,i.e.,

We next introduce the following weighted Sobolev spacesL2(Rn);j=1,2,···,n}associated with the real vector fields X=(X1,X2,···,Xn),thenis a Hilbert space endowed with normNow letΩbe a bounded open domain in Rnwith smooth boundary such thatwe denote bythe closure ofwith respect to the normWe know thatis a Hilbert space as well.

In this paper,the Dirichlet eigenvalue problem(1.1)of degenerate elliptic operator−will be considered in the weak sense innamely,

Based on assumptions(H1)–(H5)above,we can show that the Dirichlet eigenvalue problem(1.6)has a sequence of discrete eigenvalueswhich satisfy 0<λ1≤ λ2≤···≤ λk≤ ···andλk→+∞as k→+∞.

By using the regularity results of Franchi and Lanconelli[2]we can prove that the problem(1.6)has discrete Dirichlet eigenvalues.Then,by using the process of refinement in Li-Yau[4],we obtain an explicit lower bound estimates of Dirichlet eigenvaluesλkas follows.

Theorem 1.1Let X=(X1,···,Xn)be real continuous vector fields defined in Rnand satisfy assumptions(H1)–(H5).Assume thatΩis a bounded open domain in Rnwith smooth boundary such thatIf we denote byλkthe kthDirichlet eigenvalue of operator−onΩ,then for any k≥1,we have

where Q is defined by(1.4)andandΓ(x)is the Gamma function,|Ω|is the n-dimensional Lebesgue measure ofΩand C=C(X,Ω)is a positive constant.

Remark 1Sincethen Theorem 1.1 implies that the kthDirichlet eigenvalueλksatisfiesfor all k≥1.

Remark 2In general,for degenerate case we haveIf Q=n,the operator will be non-degenerate and the positive constant C can be replaced bythus estimate(1.7)will be the generalization as the Li-Yau’s lower bound estimate in[4].

Moreover,if the vector fields satisfy assumption(H6),then we have the following sharper lower bounds.

Theorem 1.2Let X=(X1,···,Xn)be real continuous vector fields defined in Rnand satisfy assumptions(H1)–(H3)and(H6).Assume thatΩis a bounded open domain in Rnwith smooth boundary such thatDenote byλkthe kthDirichlet eigenvalue of operator−onΩ,andis the homogeneous dimension of Rnwith respect to(δt)t>0.Then for any k≥1,we have

Remark 3If the vector fields admit the homogeneous structure assumptions(H1)–(H3)and(H6),then assumptions(H4)and(H5)will be also satisfied.But we cannot deduce assumption(H6)from assumptions(H1)–(H5),for example,X=(∂x1,∂x2,(|x1|α+|x2|β)∂x3)withα>β>0.

Remark 4If the vector fields admit the homogeneous structure assumptions(H1)–(H3)and(H6),then the lower bounds in(1.8)is sharper than(1.7)in the sense of growth order.

Furthermore, by the same condition in Krger [5], we obtain an upper bound for the Dirichlet eigenvalues of operator −.

Theorem 1.3Let X = (X1,...,Xn) be real continuous vector fields defined in Rnand satisfy assumptions (H1)–(H5).Suppose that Ω is a bounded open domain in Rnwith smooth boundary ∂Ω such thatMoreover, we assume that there exists a constant C0> 0 such that the measure of inner neighbourhood of the boundarysatisfies thatfor anywhereis the distance function and |Ω| is the n-dimensional Lebesgue measure of Ω.Denote by λkthe kthDirichlet eigenvalue of operator −µon Ω.Then for any k ≥, we have

Remark 5For a bounded domain Ω, if the (n − 1)-dimensional Lebesgue measure of ∂Ω is bounded, thenThus the condition in Theorem 1.3 holds for some positive constant C0.

The details of proofs for Theorem 1.1, Theorem 1.2 and Theorem 1.3 have been given by [1].