Regularity of hyperbolic equations under L2(0, T; L2(Γ))-Dirichlet boundary terms

With Ω an open bounded domain in Rn with boundary Γ, let f(t; f0, f1;u) be the solution to a second order linear hyperbolic equation defined on Ω, under the action of the forcing term u(t) applied in the Dirichlet B.C., and with initial data f0 ∈L2(Ω) and f1 ∈H-1(Ω). In a previous paper [6], we proved (among other things) that the map u → f ⊗ ft, from the Dirichlet input into the solution is continuous from L2(0, T; L2(Γ)) into L2(0, T; L2(Ω))⊗L2(0, T; H-1(Ω)). Here, we make crucial use of this result to present the following marked improvement: the map u → f ⊗ft is continuous from L2(0, T; L2(Γ)) into C([0, T]; L2(Ω))⊗C([0, T]; H-1(Ω)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the corresponding quadratic optimal control problem [7]. When u, instead, acts in the Neumann B. C. and Ω is either a sphere or a parallelepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity in t of the solution (i.e., from L2(0, T) to C[0, T]). © 1983 Springer-Verlag New York Inc