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A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations

A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain Ω with boundary Γ, under the action of a boundary forcing term inL 2(0,T; L 2(Γ)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL 2(0,T; L 2(Ω)), when Ω is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL 2(0,T; H 3/4-e(Ω)) when Ω is a parallelepiped and inL 2(0,T; H 2/3(Ω) when Ω is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations

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References (33)

Publisher
Springer Journals
Copyright
Copyright © 1981 by Springer-Verlag New York Inc.
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/BF01442108
Publisher site
See Article on Publisher Site

Abstract

This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain Ω with boundary Γ, under the action of a boundary forcing term inL 2(0,T; L 2(Γ)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL 2(0,T; L 2(Ω)), when Ω is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL 2(0,T; H 3/4-e(Ω)) when Ω is a parallelepiped and inL 2(0,T; H 2/3(Ω) when Ω is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 23, 2005

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