Engel群上Yang不等式的推广

The Promotion of Yang Inequality on the Engel Group

本文研究了Engel群上sub-Laplace算子的Dirichlet问题{-ΔEu=λu在Ω内u=0在Ω上,其中ΔE=X_1~2+X_2~2为Engel群上的sub-Laplace算子,X1,X2为Engel群上的左不变向量场.利用Chebyshev不等式及算子特征值、特征函数的性质得到了此问题特征值的不等式kΣi = 1(λk+1-λi)α≤2^(1/2)(kΣi=1(λk+1-λi)βkΣi=1(λk+1-λi)2α-β-1λi)1/2其中,α∈R,β≥0且α2≤2β.当α=β=2时即为Yang不等式,所以上述不等式是Yang不等式的一个推广.

The Dirichlet problem for sub-Laplace operator on the Engel group is explored in the paper {-△Eu=λu u=0 inΩ on αΩ Here △E=X1^2＋X2^2is the sub-Laplace operator on the Engel group and X1 ,X2 is the left invariant vector fields.And then established a inequality of eigenvalue k∑i=1（λk＋1-λi）^a≤√2（k∑i=1（λk＋1-λi）^βk∑i=1（λk＋1-λi）^2α-β-1λi）1/2 where α∈R,β≥0 and α^2≤2β by using the Chebyshev inequality and the properties of eigenvalues and eigenfunctions It is the Yang inequality when α=β=2 on the basis of such inequality. So this inequality is a promotion inequality of Yang inequality.