论文标题
完整的riemannian歧管中的热量半群的梯度估计值
Gradient estimates for the heat semigroup on forms in a complete Riemannian manifold
论文作者
论文摘要
我们研究热方程$ \ frac {\ partial u} {\ partial t}-Δu= 0，\ u（x，0）=ω（x），$ whene $δ：= dd^{*}+d^{*} d $是hodge laplacian and $（ riemannian歧管$（m，g）。$在弱小的几何假设下，我们得到对形式的半群的估计： 用$ p $ - forms用$ p \ geq 1 $和$ k \ geq 0 $： $ \ displayStyle \ forall t \ geq 1，\ {\ left \ vert {\ nabla^{k} 在功能上行动，即用$ p = 0，$我们得到更好的结果： $ \ displayStyle \ forall k \ geq 1，\ \ forall t \ geq 1，\ {\ left \ nabla^{\ nabla^{k} e^{ - tδ}} \ right \ right \ right \ right \ vert} _ { c（n，r，k）t^{ - 1/2}。$
We study the heat equation $\frac{\partial u}{\partial t}-Δu=0,\ u(x,0)=ω(x),$ where $Δ:=dd^{*}+d^{*}d$ is the Hodge laplacian and $u(\cdot ,t)$ and $ω$ are $p$-differential forms in the complete Riemannian manifold $(M,g).$ Under weak bounded geometrical assumptions we get estimates on its semigroup of the form: acting on $p$-forms with $p\geq 1$ and $k\geq 0$: $\displaystyle \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-tΔ_{p}}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k).$ Acting on functions, i.e. with $p=0,$ we get a better result: $\displaystyle \forall k\geq 1,\ \forall t\geq 1,\ {\left\Vert{\nabla ^{k}e^{-tΔ}}\right\Vert}_{L^{r}(M)-L^{r}(M)}\leq c(n,r,k)t^{-1/2}.$
