Semigroup Forum OF1–OF15 c  2003 Springer-Verlag New York Inc. DOI: 10.1007/s00233-002-0010-8 RESEARCH ARTICLE The Laplacian with Wentzell-Robin Boundary Conditions on Spaces of Continuous Functions * W. Arendt, G. Metafune, D. Pallara, and S. Romanelli Communicated by Rainer Nagel Dedicated to Jerry Goldstein on the occasion of his 60 th birthday Abstract We investigate the Laplacian ∆ on a smooth bounded open set Ω ⊂ R n with Wentzell-Robin boundary condition βu + ∂u ∂ν +∆u = 0 on the boundary Γ. Under the assumption β ∈ C(Γ) with β ≥ 0, we prove that ∆ generates a differentiable positive contraction semigroup on C( ¯ Ω) and study some mono- tonicity properties and the asymptotic behaviour. Key words: Wentzell-Robin boundary conditions, positive contraction semi- groups Mathematics subject classification (2000): 47D06, 35J20, 35J25 Introduction The aim of this article is to show that the Laplacian ∆ with Wentzell-Robin boundary condition βu + ∂u ∂ν +∆u =0onΓ (1) generates a positive contraction semigroup T on C( ¯ Ω). Here Ω is a bounded open subset of R n with smooth boundary Γ and 0 ≤ β ∈ C(Γ). Note that (1) is a dynamic boundary condition. In fact, let f be an element of C( ¯ Ω) and u(t)= T (t)f . Then u ′ (t)=∆u(t). Introducing this in (1) we obtain d dt u(t)= -βu(t) - ∂ ∂ν u(t) on Γ. We also establish monotonicity properties of this semigroup with respect to β . Also the asymptotic behaviour for t →∞ is studied. The boundary condition (1) was first studied in [9] in the space C([0, 1]) and then in [10] in the space * Work partially supported by GNAMPA-INdAM. The first author is most grateful for the hospitality of the Universities of Bari and Lecce.