Generation of Cosine Families on L ^p(0,1) by Elliptic Operators with Robin Boundary Conditions

Abstract

Let a ∈ W ^1,∞(0,1), a(x) ≥ α > 0, b, c ∈ L ^∞ (0,1) and consider the differential operator A given by Au = au″ + bu′ + cu. Let α _j , β _j (j = 0, 1) be complex numbers satisfying α _j , β _j ≠ (0,0) for j = 0, 1. We prove that a realization of A with the boundary conditions

$$ \alpha _j u\prime \left( j \right) + \beta _j u\left( j \right) = 0,{\text{ }}j = 0,1, $$

generates a cosine family on L ^p(0, 1) for every p ∈ [1, ∞]. This result is obtained by an explicit calculation, using simply d’Alembert’s formula, of the solutions in the case of the Laplace operator.

We dedicate this work to Günter Lumer in admiration