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

We consider the Laplacian R subject to Robin boundary conditions ∂u ∂ν + βu = 0 on the space C(Ω), where Ω is a smooth, bounded, open subset of RN. It is known that R generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C0(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from C(Ω) to C0(RN) which maps an operator core for R into the domain of the generator of the Gaussian semigroup. © 2009 Elsevier Inc. All rights reserved