The Robin and Wentzell-Robin Laplacians on Lipschitz domains. (English) Zbl 1168.35340

Warma, Mahamadi

Semigroup Forum 73, No. 1, 10-30 (2006).

Summary: Let \(\Omega \subset {\mathbb R}^N\) be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions \(\frac{\partial u}{\partial \nu} + \beta u = 0\) on the boundary \(\partial \Omega\) generates a holomorphic \(C_0\)-semigroup of angle \(\pi/2\) on \(C(\overline\Omega)\) if \(0 < \beta_0 \leq \beta \in L^\infty(\partial\Omega)\). With the same assumption on \(\Omega\) and assuming that \(0 \leq \beta \in L^\infty(\partial\Omega)\), we show in the second part that one can define a realization of the Laplacian on \(C(\overline\Omega)\) with Wentzell-Robin boundary conditions \(\Delta u + \frac{\partial u}{\partial \nu} + \beta u=0\) on the boundary \(\partial\Omega\) and this operator generates a \(C_0\)-semigroup.

MSC:

35J25	Boundary value problems for second-order elliptic equations
35J20	Variational methods for second-order elliptic equations
47D03	Groups and semigroups of linear operators