I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ such that \begin{equation} p(x,y,t)\geq \frac{1}{Ct^{n/2}} e^{-\frac{Cd(x,y)^2}{t}} \end{equation} where $d(x,y)$ is the geodesic distance in $U$ between $x$ and $y$.
I could not find any references that gives the theorem in the form stated above. Other similar results I've found until now and the reasons why the do not satisfy my needs are listed below.
- In [1] the result is given for a complete Riemannian manifold.
- In [2] the estimate near the boundary is worser.
- In [3] the result is given for convex domains.
- In [4] the result is given for inner uniform domains.
- In [5] the result is given for the Dirichlet heat kernel on inner uniform domains.
Furthermore, in [5] the inner uniform domains considered are assumed to have a smooth boundary, but this has not been proven. So if we know the exact proof of this, We could use the result of [4] by changing from smooth to $C^2$. Therefore, I would like to know not only where I can find a proof of the above inequality but also matters related to its application on an inner uniform domain.
Thanks.
References:
[1] Sharp explicit lower bounds of heat kernels, by Feng-Yu Wang.
[2] Heat kernel estimates for general boundary problems, by Liangpan Li and Alexander Strohmaier.
[3] On domain monotonicity of the Neumann heat kernel, by Richard F. Bass and Krzysztof Burdzy.
[4] The heat kernel and its estimates, by Laurent Saloff-Coste.
[5] The Dirichlet heat kernel in inner uniform domains: Local
results, compact domains and non-symmetric forms, by Janna Lierl and Laurent Saloff-Coste.
