摘要
Let e_λ(x) be a Neumann eigenfunction with respect to the positive Laplacian A on a compact Riemannian manifold M with boundary such that △e_λ=λ~2e_λ in the interior of M and the normal derivative of e\ vanishes on the boundary of M.Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M.The authors show the following gradient estimate for χλf as λ≥1:‖▽χλ f‖∞≤C(λ‖χλ f‖∞+λ^(-1)‖△χλf‖∞),where C is a positive constant depending only on M,As a corollary,the authors obtain the gradient estimate of e_λ:For every λ≥1,it holds that ‖▽e_λ‖∞≤Cλ‖e_λ‖∞.
Let eλ(x) be a Neumann eigenfunction with respect to the positive Laplacian λ on a compact Riemannian manifold M with boundary such that A eλ = λ2eλ in the interior of M and the normal derivative of ex vanishes on the boundary of M. Let xλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate
基金
supported by the National Natural Science Foundation of China(Nos.10971104,11271343,11101387)
the Anhui Provincial Natural Science Foundation(No.1208085MA01)
the Fundamental Research Funds for the Central Universities(Nos.WK0010000020,WK0010000023,WK3470000003)