The heat kernel in a domain with Dirichlet boundary conditions satisfies so-called “not feeling the boundary estimates”. These reflect the locality of the heat expansion and can for example be derived using Brownian motion. Very precise explicit estimates can be found in the literature. These estimates and the methods used to prove them do not automatically generalize to other boundary conditions.
I will show that finite propagation speed estimates and Fourier Tauberian theorems can be used to prove such estimates for general boundary conditions.
(joint work with Liangpan Li)