A cosine operator approach to modeling L2(0, T; L2 (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 in L2(0, T; L2(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 in L2(0, T; L2(Ω)), when Ω is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is in L2(0, T; H3/4-e(Ω)) when Ω is a parallelepiped and in L2(0, T; H2/3(Ω) when Ω is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques. © 1981 Springer-Verlag New York Inc.

Publication Title

Applied Mathematics & Optimization