华东师范大学学报(自然科学版) ›› 2010, Vol. 2010 ›› Issue (1): 91-98.

• 应用数学与基础数学 • 上一篇    下一篇

关于 p-Laplace 算子的一个最优估计

王林峰   

  1. 南通大学 理学院, 江苏 南通226000
  • 收稿日期:2009-02-02 修回日期:2009-05-07 出版日期:2010-01-25 发布日期:2010-01-25
  • 通讯作者: 王林峰

Sharp estimate of the p-Laplace operator

WANG Lin-feng   

  1. School of Science, Nantong University, Nantong Jiangsu226000, China
  • Received:2009-02-02 Revised:2009-05-07 Online:2010-01-25 Published:2010-01-25
  • Contact: WANG Lin-feng

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摘要： M是带度量 g 的 n 维非紧黎曼流形,1<pleqslant 2 给定常数,triangle_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triangle_pu=-lambda|u|^p-2u 的正函数的一个梯度估计, 其中 lambdageqslant 0是常数; 同时得到了lambda 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triangle 的椭圆方程 triangle u=-lambda u 梯度估计的结果.

关键词: p-Laplace, 梯度估计, 最优估计, p-Laplace, 梯度估计, 最优估计

Abstract: 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.

Key words: gradient estimate, sharp estimate, p-Laplace operator, gradient estimate, sharp estimate

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