The wave equation with Wentzell–Robin boundary conditions on Lp-spaces

Let a∈W1,∞(0,1), a(x)⩾δ>0, b,c∈L∞(0,1) and consider the differential operator A formally given by Au=au″+bu′+cu. We prove in the first part that a realization of A with Wentzell–Robin boundary conditions on Lp(0,1)×C2 generates a cosine function for p∈[1,∞). In particular, we obtain that this realization of A generates a holomorphic C0-semigroup of angle π/2 on the space L1(0,1)×C2. This solves an open problem by A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli and W. Arendt. Of crucial importance is the formulation of the boundary conditions. We show in the second part that if Ω:=N(0,1), N⩾1, is the cube in RN, then the Laplacian with pure Wentzell boundary conditions generates an α-times integrated cosine function on for any .