Analysis of the heat kernel of the Dirichlet-to-Neumann operator

TL;DR

This paper establishes Poisson upper bounds for the heat kernel associated with the Dirichlet-to-Neumann operator on smooth bounded domains, extending bounds to complex parameters and derivatives.

Contribution

It provides new Poisson bounds for the heat kernel of the Dirichlet-to-Neumann operator, including complex and derivative bounds, in smooth bounded domains.

Findings

Poisson upper bounds for the kernel $K$ on smooth bounded domains

Poisson bounds for $K_z$ in the right half-plane

Bounds extend to derivatives of the kernel

Abstract

We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the right half-plane and for all its derivatives.