\(L^ p\) regularity for the wave equation with strictly convex obstacles (Q1325153)

The authors prove estimates in \(L^ p\)-spaces for solutions of the wave equation in \(\mathbb{R}^ n\) with a strictly convex obstacle \(\Omega\). The proofs base on \(H^ 1 - L^ 1\) continuity results for Fourier-Airy integral operators using certain oscillatory integral representations. They find estimates for the operator \(\exp (it \sqrt {- \Delta})\) in Dirichlet-Sobolev spaces \[ L^ p_ s (\mathbb{R}^ n \backslash \Omega) = (1 - \Delta)^{-s/2} (L^ p (\mathbb{R}^ n \backslash \Omega) ) \] which are similar to the case without obstacle. Here \(\Delta\) denotes the Laplace operator on the exterior \(R^ n \backslash \Omega\) with Dirichlet boundary conditions. This implies \(L^ p\)-regularity results for the corresponding Cauchy problem with obstacle \(\Omega\). They also get \(L^ p\)-estimates for solutions to the Dirichlet problem which are uniform close to \(\partial \Omega\). Especially this implies good boundary behaviour of the solutions in \(L^ p\)-spaces. Moreover they treat \(L^ p\)-mapping properties of certain Fourier integral operators associated to folding canonical relations.