On the diffraction of waves by conical singularities. II. (English) Zbl 0536.58032

This paper concludes the discussion begun in the first part [ibid. 25, 275-331 (1982; Zbl 0526.58049)]. Let \(N^ m\) be a Riemannian manifold. The metric cone \(C(N^ m)\), on \(N^ m\), is the space \({\mathbb{R}}^+\times N^ m\) together with the Riemannian metric \(dr^ 2+r^ 2\tilde g,\) where \(\tilde g\) is the Riemannian metric on \(N^ m\). The Laplacian on functions on \(C(N^ m)\) is \[ \Delta =-\frac{\partial^ 2}{\partial r^ 2}-\frac{m}{r}\frac{\partial}{\partial r}+r^{-2}{\tilde \Delta} \] where \({\tilde \Delta}\) denotes the Laplace operator on the base \(N^ m\). The paper studies solutions to the wave equation on manifolds with conical singularities, particularly subsets of Euclidean space with singularities of this type, by using a functional calculus for \(\Delta\) on \(C(N^ m)\). A principal object of the paper is to study singularities of the kernel of the wave kernel \(\sin \sqrt{\Delta t}/\sqrt{\Delta},\) which may be expressed in terms of the pseudodifferential operator \(\nu \equiv({\tilde \Delta}+\alpha^ 2)^{1/2}\), where \(\alpha \equiv(1- m)/2\). Part I comprises sections. Section 1 discusses functions of \(\nu\) ; section 2 gives explicit constructions of some functions of \(\nu\) when \(N=S^ m\), the unit sphere, using a method of descent, which is generalized in section 4.5 of part II. Section 3 summarizes some basic results for the Laplacian on spaces with conical singularities, reviews the construction of \(\sin \sqrt{\Delta t}/\sqrt{\Delta}\) and describes its wavefront set, and then proves the basic result on propagation of singularities, for arbitrary solutions of the wave equation. The present paper (part II) continues with sections 4, 5, 6. Section 4 illustrates the results of section 3 by applying them to \(N=S^ 1_ 2\) and \(S^ 1_{1/\gamma},\) the circles of circumference 4\(\pi\) and \(2\pi /\gamma\) resp., right circular cones in \({\mathbb{R}}^ 4\) and \({\mathbb{R}}^ 3\) with source on the axis and Dirichlet boundary conitions, and then treats polyhedral corners in \({\mathbb{R}}^ n\), pseudomanifolds and suspension. Explicit solutions are obtained via expressions for cos \(\nu\) s and other functions of \(\nu\). Section 5 discusses the asymptotic behaviour of the fundamental solution near the diffracted wavefront. A complete asymptotic expansion is obtained for the solution even near the ’forward’ directions, which are particularly complicated. Section 6 proves local exponential energy decay for the exterior of a convex region with conical points in \({\mathbb{R}}^{2k+1}\).
Reviewer: F.J.Wright


58J45 Hyperbolic equations on manifolds
35L05 Wave equation


Zbl 0526.58049
Full Text: DOI


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