The wave equation with Wentzell–Robin boundary conditions on [formula omitted]-spaces

Abstract

Let a ∈ W 1 , ∞ ( 0 , 1 ) , a ( x ) ⩾ δ > 0 , b , c ∈ L ∞ ( 0 , 1 ) and consider the differential operator A formally given by A u = a u ″ + b u ′ + c u . We prove in the first part that a realization of A with Wentzell–Robin boundary conditions on L p ( 0 , 1 ) × C 2 generates a cosine function for p ∈ [ 1 , ∞ ) . In particular, we obtain that this realization of A generates a holomorphic C 0 -semigroup of angle π / 2 on the space L 1 ( 0 , 1 ) × C 2 . 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 : = ( 0 , 1 ) N , N ⩾ 1 , is the cube in R N , then the Laplacian with pure Wentzell boundary conditions generates an α-times integrated cosine function on C ( Ω ¯ ) for any α ⩾ ( N − 1 ) 2 .