完备黎曼流形上微分形式上热半群的梯度估计
偏微分方程分析
2022-07-01 v2
摘要
我们研究热方程 ∂ u ∂ t − Δ u = 0 , u ( x , 0 ) = ω ( x ) , \frac{\partial u}{\partial t}-\Delta u=0,\ u(x,0)=\omega (x), ∂ t ∂ u − Δ u = 0 , u ( x , 0 ) = ω ( x ) , Δ : = d d ∗ + d ∗ d \Delta :=dd^{*}+d^{*}d Δ := d d ∗ + d ∗ d u ( ⋅ , t ) u(\cdot ,t) u ( ⋅ , t ) ω \omega ω ( M , g ) (M,g) ( M , g ) p p p p ≥ 1 p\geq 1 p ≥ 1 k ≥ 0 k\geq 0 k ≥ 0 p p p ∀ t ≥ 1 , ∥ ∇ k e − t Δ p ∥ L r ( M ) − L r ( M ) ≤ c ( n , r , k ) . \displaystyle \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-t\Delta_{p}}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k). ∀ t ≥ 1 , ∇ k e − t Δ p L r ( M ) − L r ( M ) ≤ c ( n , r , k ) . p = 0 p=0 p = 0 ∀ k ≥ 1 , ∀ t ≥ 1 , ∥ ∇ k e − t Δ ∥ L r ( M ) − L r ( M ) ≤ c ( n , r , k ) t − 1 / 2 . \displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-t\Delta }}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k)t^{-1/2}. ∀ k ≥ 1 , ∀ t ≥ 1 , ∇ k e − t Δ L r ( M ) − L r ( M ) ≤ c ( n , r , k ) t − 1/2 .
引用
@article{arxiv.2003.03985,
title = {Gradient estimates for the heat semigroup on forms in a complete Riemannian manifold},
author = {Eric Amar},
journal= {arXiv preprint arXiv:2003.03985},
year = {2022}
}
备注
we correct some mistakes and modify the presentation