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

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
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