The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions
The authors study the Laplacian \(\Delta\) on a smooth bounded domain \(\Omega\subset {\mathbb R}^n\) with Wentzell-Robin boundary condition \(\beta u +\frac{\partial u}{\partial \nu} +\Delta u=0\) on the boundary \(\Gamma\). The aim is to show that the operator generates a positive differentiable contraction semigroup on \(C(\bar\Omega)\). For this, the authors use the method of quadratic forms to construct a semigroup on \(L^2(\Omega)\oplus L^2(\Gamma)\) and show that it is a Markov semigroup, thus extendable to \(L^p(\Omega)\oplus L^p(\Gamma),\;1\leq p<\infty\). By means of Schauder estimates it is then proved that the semigroup is a Feller semigroup on \(C(\bar\Omega)\). Some monotonicity properties and the asymptotic behaviour are established.
scientific article; zbMATH DE number 2051543