Approximation of the Semigroup generated by the Robin Laplacian in terms of the Gaussian Semigroup

TL;DR

The paper develops an approximation method for the semigroup generated by the Robin Laplacian using extension operators, providing a Trotter-like formula and addressing Dirichlet limits.

Contribution

It introduces contractive extension operators for Robin Laplacian domains and establishes a Trotter-like approximation for the associated semigroup.

Findings

Constructed extension operators preserving regularity.

Proved a Trotter-like approximation for Robin Laplacian semigroup.

Addressed the Dirichlet boundary condition case separately.

Abstract

For smooth bounded open sets in euclidean space, we construct corresponding contractive linear extension operators for the space of continuous functions which preserve regularity of functions in the domain of the Robin Laplacian. We also prove a Trotter-like approximation for the semigroup generated by the Laplacian subject to Robin boundary conditions in terms of these extension operators. The limiting case of Dirichlet boundary conditions is treated separately.