Propagation of singularities for the wave equation on edge manifolds

Melrose, Richard; Vasy, András; Wunsch, Jared

doi:10.1215/00127094-2008-033

Cited by 42 publications

(149 citation statements)

References 21 publications

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“…[1], [10]. The present work (and ongoing projects continuing it, especially joint work with Melrose and Wunsch [15], see also [2], [16]), can be considered a justification of Keller's work in the general geometric setting (curved edges, variable coefficient metrics, etc. ).…”

Section: Our Main Results Ismentioning

confidence: 82%

“…In the setting of (isolated) conic points, such a result was obtained by Cheeger, Taylor, Melrose and Wunsch [2], [16]. While the analogous result (including its precise statement) for manifolds with corners is still some time away, significant progress has been made, since the original version of this manuscript was written, on analyzing edge-type metrics (on manifolds with boundaries) in the project [15]. The outline of these results, including a discussion of how it relates to the problem under consideration here, is written up in the lecture notes of the author on the present paper [26].…”

Section: Our Main Results Ismentioning

confidence: 84%

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Propagation of singularities for the wave equation on manifolds with corners

Vasy

2008

Ann. Math.

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219

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In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M × R t , P = D 2 t − ∆ M , and u ∈ H 1 loc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

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Section: Our Main Results Ismentioning

confidence: 82%

Section: Our Main Results Ismentioning

confidence: 84%

Propagation of singularities for the wave equation on manifolds with corners

Vasy

2008

Ann. Math.

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219

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“…However, we can still define a flowout map from the boundary: employing a variant on the notation of [MVW08], we denote the "flowout map"…”

Section: Conic Geometrymentioning

confidence: 99%

The diffractive wave trace on manifolds with conic singularities

Ford¹,

Wunsch²

2017

Advances in Mathematics

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Abstract. Let (X, g) be a compact manifold with conic singularities. Taking ∆g to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group e −it √ ∆g arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points. IntroductionIn this paper, we consider the trace of the half-wave group U(t) def = e −it √ ∆g on a compact manifold with conic singularities (X, g). Our main result is a description of the singularities of this trace at the lengths of closed geodesics undergoing diffractive interaction with the cone points. Under the generic assumption that the cone points of X are pairwise nonconjugate along the geodesic flow, the resulting singularity at such a length t = L has the oscillatory integral representationwhere the amplitude a is to leading orderas |ξ| → ∞ and the index j is cyclic in {1, . . . , k}. Here, n is the dimension of X and k the number of diffractions along the geodesic, and χ is a smooth function supported in [1, ∞) and equal to 1 on [2, ∞). L 0 is the primitive length, in case the geodesic is an iterate of a shorter closed geodesic. The product is over the diffractions undergone by the geodesic, with D αj a quantity determined by the functional calculus of the Laplacian on the link of the j-th cone point Y αj , the factor Θ − 1 2 (Y αj → Y αj+1 ) is (at least on a formal level) the determinant of the differential of the flow between the j-th and (j + 1)-st cone points, and m γj is the Morse index of the geodesic segment γ j from the j-th to (j + 1)-st cone points. All of these factors are described in more detail below.To give this result some context, we recall the known results for the Laplace- is a well-defined distribution on R t . Moreover, it satisfies the "Poisson relation": it is singular only on the length spectrum of (X, g),L is the length of a closed geodesic in (X, g)} .(This was shown independently by Chazarain [Cha74]; see also [CdV73].) Subject to a nondegeneracy hypothesis, the singularity at the length t = ±L of a closed geodesic has a specific leading form encoding the geometry of that geodesic-the formula involves the linearized Poincaré map and the Morse index of the geodesic. The proofs of these statements center around the identificationwhere∆g is (half of) the propagator for solutions to the wave equation on X; in particular, U(t) is a Fourier integral operator.In this paper, we prove an analogue of the Duistermaat-Guillemin trace formula on compact manifolds with conic singularities (or "conic manifolds"), generalizing results of Hillairet [Hil05] from the case of flat surfaces with conic singularities. We again consider the trace Tr U(t), a spectral invariant equal to ∞ j=0 e −itλj . The Poiss...

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“…(See section 3.1 and [12, Lemma 3.8] for more details.) The more general arguments of Vainberg rely on results about propagation of singularities, and the relevant results for nonsmooth domains have only recently been obtained (see [40], [38], [59], [39], [8], and section 3.1).…”

Section: Introduction Proving Bounds On Solutions Of the Helmholtz Ementioning

confidence: 99%

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

Spence¹

2014

SIAM J. Math. Anal.

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Abstract. We prove wavenumber-explicit bounds on the Dirichlet-to-Neumann map for the Helmholtz equation in the exterior of a bounded obstacle when one of the following three conditions holds: (i) the exterior of the obstacle is smooth and nontrapping, (ii) the obstacle is a nontrapping polygon, or (iii) the obstacle is star-shaped and Lipschitz. We prove bounds on the Neumann-toDirichlet map when condition (i) and (ii) hold. We also prove bounds on the solutions of the interior and exterior impedance problems when the obstacle is a general Lipschitz domain. These bounds are the sharpest yet obtained (for their respective problems) in terms of their dependence on the wavenumber. One motivation for proving these collection of bounds is that they can then be used to prove wavenumber-explicit bounds on the inverses of the standard second-kind integral operators used to solve the exterior Dirichlet, Neumann, and impedance problems for the Helmholtz equation.

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Propagation of singularities for the wave equation on edge manifolds

Cited by 42 publications

References 21 publications

Propagation of singularities for the wave equation on manifolds with corners

Propagation of singularities for the wave equation on manifolds with corners

The diffractive wave trace on manifolds with conic singularities

Wavenumber-Explicit Bounds in Time-Harmonic Acoustic Scattering

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