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...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展开更多
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)
文摘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
