关于p-Laplace算子的一个最优估计

M是带度量g的n维非紧黎曼流形,1p≤2给定常数,△_p是M上的p-Laplace算子,借助于经典的Li-Yau的方法证明了在一定的曲率条件下,满足方程△_pu=-λ|u|~(p-2)u的正函数的一个梯度估计,其中λ≥0是常数;同时得到了λ的一个上界估计;进一步说明了此估计是最优的.推广了关于Laplace算子△的椭圆方程△u=-λu梯度估计的结果.

Sharp estimate of the p-Laplace operator

Let M be an n-dimensional complete noncompact Riemannian manifold with metric g, triangle_p(1<pleqslant 2) the p-Laplace operator, by using the classical method of Li-Yau, a gradient estimate of the positive solution to equation triangle_pu=-lambda |u|^p-2u was proved under suitable curvature condition, in which lambdageqslant 0 is a constant; the upper bound estimate of lambda was a byproduct; one also showed that this estimate is sharp. This result generalizes the gradient estimate of the positive solution to elliptic equation triangle u=-lambda u.