I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann heat kernels (for the Laplacian operator) in exterior domains (i.e. $\Omega=\mathbb{R}^N\backslash K$ where $K$ is a regular compact set).

I have found multiple results from Saloff-Coste, Grigor'yan, Davies, Q. S. Zhang... (See the references below). Most of them concern the problem in complete Riemann manifolds. Some of them study the problem in exterior domains (or in more general (if $K$ is regular enough) inner uniform domains) [2,4,5]. A good general text about this is this one from Saloff-Coste: [1].

However, most of the results I have found are bounds like the following: $$ \phi(x,y,t)\frac{ e^{-c_1\frac{d(x,y)^2}{4t}}}{(4\pi t)^{N/2}}\leq k(x,y,t)\leq \Phi(x,y,t)\frac{ e^{-c_2\frac{d(x,y)^2}{4t}}}{(4\pi t)^{N/2}} $$ where $d(x,y)$ is the geodesic distance in $\Omega$ between $x$ and $y$ and $\phi,\Phi$ are known functions. See for example [2] Th 1.1. and [4] Th 1.3.1, 1.3.3.

I would like to know if there are some sharp results in which $c_1=1$ and/or $c_2=1$, like the heat kernel in the whole space. I have not found almost any result in this way. One I found is [6] Cor 3.1, Th 4.1 in which they get $c_1=1+\varepsilon$ and $c_2=1-\varepsilon$ with $\varepsilon$ as small as one would like (but then $\phi$ and $\Phi$ change obviously) but it is for complete manifolds.

Any comment, suggestion or reference related to this would be really helpful. In particular my main questions are:

- Are there any sharp result of that type? It is useful for me even if it is not the case of exterior domain, but other settings.
- Are there any result in which I can make $c_1$ and $c_2$ as close to $1$ as we want, although not $1$?
- If you don't know about the first two questions, do you think it is a plausible result to obtain that type of bounds for the Dirichlet Heat Kernel in exterior domains?

I am also interested in Neumann heat kernel, so any results for it are also welcomed.

References:

[1] The heat kernel and its estimates. L. Saloff Coste.

[2] The global behavior of heat kernels in exterior domains. Q. S. Zhang.

[3] Heat Kernels and Spectral Theory. E. B. Davies.

[4] Heat kernel estimates for inner uniform subsets of Harnack-type Dirichlet space. P. Gyrya.

[5] Dirichlet heat kernel in the exterior of a compact set Grigor'yan, Saloff-Coste.

[6] On the parabolic kernel of the Schrödinger operator P. Li, S. T. Yau

PS: This is my first question here, do not hesitate to correct me if the question is not well formulated.