Abstract
Let \(\Omega\subset{\Bbb R}^N\) 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 \(0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)\). With the same assumption on \(\Omega\) and assuming that \(0\le\beta\in L^{\infty}(\partial \Omega)\), 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.
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Warma, M. The Robin and Wentzell-Robin Laplacians on Lipschitz Domains. Semigroup Forum 73, 10–30 (2006). https://doi.org/10.1007/s00233-006-0617-2
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DOI: https://doi.org/10.1007/s00233-006-0617-2