Abstract
Using the abstract framework [Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations with generalized Wentzell boundary conditions. Journal of Differential Equations, 207, 1–20.] we show that certain second-order differential operators with generalized Wentzell boundary conditions generate cosine families and hence also analytic semigroups on W1,1(0,1). This complements the main result [Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht, E. and Romanelli, S., 2003, General Wentzell boundary conditions and analytic semigroups on W1, p (0,1). Applicable Analysis, 82, 927–935.] on the generation of an analytic semigroup by the second derivative with generalized Wentzell boundary conditions on W1, p (0, 1) for 1<p<∞.
Notes
Here [D(A)]:=(D(A),‖·‖ A ). Moreover, “X↪Y” denotes the continuous imbedding of X in Y.
Here ⟨ ··· ⟩ denotes the linear span.
Here we identify a function in