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

Abstract

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