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 domain??Rn,with smooth boundary??,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 let?be 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 that?is a bounded open domain in Rnwith smooth boundary such thatIf we denote byλkthe kthDirichlet eigenvalue of operator?on?,then for any k≥1,we have

where Q is defined by(1.4)andandΓ(x)is the Gamma function,|?|is the n-dimensional Lebesgue measure of?and C=C(X,?)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 that?is a bounded open domain in Rnwith smooth boundary such thatDenote byλkthe kthDirichlet eigenvalue of operator?on?,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 ? is a bounded open domain in Rnwith smooth boundary ?? 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 |?| is the n-dimensional Lebesgue measure of ?.Denote by λkthe kthDirichlet eigenvalue of operator ?μon ?.Then for any k ≥, we have

Remark 5For a bounded domain ?, if the (n ? 1)-dimensional Lebesgue measure of ?? 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].