Heat Kernel estimates for general boundary problems

We show that not feeling the boundary estimates for heat kernels hold for any non-negative self-adjoint extension of the Laplace operator acting on vector-valued compactly supported functions on a domain in ${\mathbb{R}}^d$. They are therefore valid for any choice of boundary condition and we show that the implied constants can be chosen independent of the self-adjoint extension. The method of proof is very general and is based on finite propagation speed estimates and explicit Fourier Tauberian theorems obtained by Y. Safarov.