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

In: Functional Analysis and Evolution Equations

Author

Listed:
  • 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))

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.

Suggested Citation

  • Ralph Chill & Valentin Keyantuo & Mahamadi Warma, 2007. "Generation of Cosine Families on L p (0,1) by Elliptic Operators with Robin Boundary Conditions," Springer Books, in: Herbert Amann & Wolfgang Arendt & Matthias Hieber & Frank M. Neubrander & Serge Nicaise & Joachim vo (ed.), Functional Analysis and Evolution Equations, pages 113-130, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7643-7794-6_7
    DOI: 10.1007/978-3-7643-7794-6_7
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