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

Ralph Chill (), Valentin Keyantuo () and Mahamadi Warma ()
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Ralph Chill: Université Paul Verlaine — Metz et CNRS, Laboratoire de Mathématiques et Applications de Metz
Valentin Keyantuo: University of Puerto Rico, Department of Mathematics (Rio Piedras Campus)
Mahamadi Warma: University of Puerto Rico, Department of Mathematics (Rio Piedras Campus)

A chapter in Functional Analysis and Evolution Equations, 2007, pp 113-130 from Springer

Abstract: 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.

Keywords: Wave equation; cosine function; second-order elliptic operators; Robin boundary conditions (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-7794-6_7

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DOI: 10.1007/978-3-7643-7794-6_7

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