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This is a similar question to Heat Kernel Asymptotics on Manifold with Boundary. But I have some further questions. Let $(M,g)$ be a closed Riemannian manifold, the heat kernel $p(x,y,t)$ of Laplace-...

Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n+1}$ with smooth boundary. $\Omega\cap\{(0,\cdots,0,y)\in\mathbb{R}^{n+1}| y\in \mathbb{R}\}\neq \varnothing$ Consider the sub-elliptic operator ...