# Sharp Dirichlet heat kernel estimates in exterior domains?

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:

1. 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.
2. Are there any result in which I can make $$c_1$$ and $$c_2$$ as close to $$1$$ as we want, although not $$1$$?
3. 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.

[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.

The upper bound with $$c_2=1$$ and a polynomial $$\Phi$$ should be in "Coulhon-Sikora: Gaussian upper bounds via Phragmen-Lindelof principle", Proceedings London Math. Soc. 96 (2008), 507-544. The setting is quite general and should cover exterior domains, too.