Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary


Let Ω ⊂Rn be a smooth domain with boundary Γ. Let u be the solution of a second order hyperbolic scalar equation with homogeneous Neumann boundary conditions defined on Ω due to initial conditions u0, u1 and to a forcing term f{hook}. The present paper studies the regularity of the map {f{hook}, u0, u1} → u|Γ, where {f{hook}, u0, u1} ε{lunate} L2(Q) × H1(Ω) × L2(Ω), Q = (0, T] × Ω, and u|Γ is the Dirichlet trace of the solution. In particular, the case of compactly supported data {f{hook}, u0, u1} is considered and contrasted with corresponding results where the data do not have compact support. In this latter case, an optimal regularity result is given for the wave equation on the half-space of dimension ≥2. It states, contrary to expectations from elliptic or parabolic theory, that u|Σ ε{lunate} H 3 4(∑), but u|Σ ∉ H 3 4 + ε(∑), ∀ε > 0, Where ∑ = (0, T] × Γ. © 1989.

Publication Title

Journal of Mathematical Analysis and Applications