Cosine families generated by second-order differential operators on W^1,1(0, 1) with generalized Wentzell boundary conditions

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 W^1,1(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 W^{1, 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 W^{1, p} (0, 1) for 1<p<∞.

Keywords:

• Sine and cosine families
• Second-order differential operators
• Wentzell boundary conditions
• Phase spaces
• 1991 Mathematics Subject Classifications: 47D09
• 34G10
• 35L05

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

with its restriction to [−1, 1].