TheNeumann Laplacian on spaces of continuous functions

Abstract

If $\Omega \subset \mathbb{R}^N$ is an open set, one can always define the Laplacian with Neumann boundary conditions $\Delta^N_\Omega$ on $L^2(\Omega)$. It is a self-adjoint operator generating a $C_O$-semigroup on $L^2(\Omega)$. Considering the part $\Delta^N_\Omega,c$ of $\Delta^N_\Omega$ in $C(\overline{\Omega})$,we ask under which conditions on it generates a $C_O$-semigroup.