带权Laplacian方程解的最优梯度估计

Sharp Gradient Estimate for Positive Solutions of the Laplacian with Drift

设M为n维完备无边界的流形,它的Ricci曲率有下界-K,这里K为实常数.假设M上的向量场B满足|B|≤γ且▽B≤K_*,这里γ为非负常数,K_*为实常数,则带权Laplacian方程ΔAu+Bu=0任意正的光滑解满足最优梯度估计|▽u|~2/(u^2)≤m(K+K_*)+((mγ~2)/(m-n)),其中任意常数m>n.

Let M be any n-dimensional complete manifold without boundary and with Ricci curvature bounded below by -K, where K is a real constant. If B is a vector field such that the norm ｜B｜≤γ and △↓B≤K＊ on M, for nonnegative constant γ and real constant K＊, then any positive smooth solution of the equation △u＋Bu =0 satisfies the following sharp gradient estimate ｜△↓u｜^2/u^2≤m（K＋K＊）＋mγ^2/m-n, on M for any real constant m 〉 n.