The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions. (English) Zbl 1073.47045

Arendt, W.; Metafune, G.; Pallara, D.; Romanelli, S.

Semigroup Forum 67, No. 2, 247-261 (2003).

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.

Reviewer: V. A. Liskevich (Bristol)

MSC:

47D06	One-parameter semigroups and linear evolution equations
35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Laplacian; Wentzell-Robin boundary condition; method of quadratic forms; Schauder estimates; monotonicity; asymptotic behaviour