Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds
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DOI：
10.1112/s0024609306018947
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发表时间：
2005-02
影响因子：
0.9
通讯作者：
P. Souplet;Qi S. Zhang
中科院分区：
文献类型：
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作者：
P. Souplet;Qi S. Zhang
我们得到一个尖锐的，局部化的版本的椭圆型梯度估计的正解（有界或无界）的热方程。这些估计与紧流形上的拉普拉斯方程的Cheng-Yau估计和热方程有界解的汉密尔顿估计有关。作为应用，我们推广丘的著名的刘维定理的正调和函数的积极的古代（包括永恒）解决方案的热方程，在一定的增长条件。令人惊讶的是，这个热方程的刘维尔定理即使在Rn中没有这样的条件也不成立。我们还证明了非紧流形上热核对数的锐化长时间梯度估计。2000年数学学科分类35 K 05、58 J35。
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng–Yau estimate for the Laplace equation and Hamilton's estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau's celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in Rn without such a condition. We also prove a sharpened long‐time gradient estimate for the log of the heat kernel on noncompact manifolds. 2000 Mathematics Subject Classification 35K05, 58J35.