Heat Kernel estimates for general boundary problems

TL;DR

This paper establishes boundary-independent heat kernel estimates for a broad class of self-adjoint Laplace operators on vector-valued functions, using general methods based on finite propagation speed and Fourier Tauberian theorems.

Contribution

It proves boundary-independent heat kernel estimates for all non-negative self-adjoint extensions of the Laplacian, regardless of boundary conditions, with constants independent of the extension.

Findings

Boundary estimates hold for all non-negative self-adjoint extensions.

Constants in estimates are independent of the boundary conditions.

Method applies to vector-valued functions on domains in Euclidean space.

Abstract

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.