Eigenvalue Problems And Gradient Estimates For Elliptic Equations On Riemannian Manifolds

In this paper，we study the eigenvalue problems and a pointwise gradient estimation forelliptic equations on Riemannian manifolds with modified Ricci curvature．Limoncu[11]introduced a new modified Ricci tensorRic kin2009and obtained a estimate of theeigenvalue of Laplacian．By the Bochner formula and integration by parts，we show that somesimilar conclusions on Riemannnian manifold with modified Ricci tensor．First，we prove that whenk1n，the result of Limoncu about the eigenvalue estimateis sharp．we also obtain a sharp estimate for the first nonzero Dirichlet or Neumanneigenvalue for the Laplacian on the manifold with a smooth boundary M．Second，we infer a rigidity theorem about eigenvalue problems on the manifold with themodified Ricci tensor．We remove the condition of the constant scalar curvature in Theorem4of Deshmukh[21]and generalize his result to the manifold with the modified Ricci tensors andobtain the same conclusion．Finally, we know that Farina and Valdinoci[23]prove a pointwise gradient bound for aquasiliear elliptic equation on compact Riemannian manifolds with nonnegative Riccicurvature in2011．We use the “P-function technique”and Maximum Principle to generalizethis result to the manifold with the Bakry-Emery Ricci curvature and obtain the optimalpointwise gradient estimation for solutions．...