## Duke Mathematical Journal

### Propagation of singularities for the wave equation on edge manifolds

#### Abstract

We investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities. This class of manifolds includes, and is modeled on, the product of a smooth manifold and a cone over a compact fiber. Our main results are a general diffractive theorem showing that the spreading of singularities at the edge only occurs along the fibers and a more refined geometric theorem showing that for appropriately regular (nonfocusing) solutions, the main singularities can only propagate along geometrically determined rays. Thus, for the fundamental solution with initial pole sufficiently close to the edge, we are able to show that the regularity of the diffracted front is greater than that of the incident wave

#### Article information

Source
Duke Math. J., Volume 144, Number 1 (2008), 109-193.

Dates
First available in Project Euclid: 2 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1215032812

Digital Object Identifier
doi:10.1215/00127094-2008-033

Mathematical Reviews number (MathSciNet)
MR2429323

Zentralblatt MATH identifier
1147.58029

#### Citation

Melrose, Richard; Vasy, András; Wunsch, Jared. Propagation of singularities for the wave equation on edge manifolds. Duke Math. J. 144 (2008), no. 1, 109--193. doi:10.1215/00127094-2008-033. https://projecteuclid.org/euclid.dmj/1215032812

#### References

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• --. --. --. --., On the diffraction of waves by conical singularities, II, Comm. Pure Appl. Math. 35 (1982), 487--529.
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• A. Hassell, R. Melrose, and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math. 181 (2004), 1--87.
• —, Scattering for symbolic potentials of order zero and microlocal propagation near radial points, preprint,\arxivmath/0502398v2[math.AP]
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• —, Singularities of boundary value problems, II, Comm. Pure Appl. Math. 35 (1982), 129--168.
• R. B. Melrose and J. Wunsch, Propagation of singularities for the wave equation on conic manifolds, Invent. Math. 156 (2004), 235--299.
• J. SjöStrand, Propagation of analytic singularities for second order Dirichlet problems, Comm. Partial Differential Equations 5 (1980), 41--93.
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• A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896), 317--374.
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• V. A. Borovikov, The Green's function for a diffraction problem on a polyhedral angle (in Russian) Dokl. Akad. Nauk SSSR 151 (1963), 251--254.
• Jeff Cheeger and Michael Taylor, On the diffraction of waves by conical singularities, I, Comm. Pure Appl. Math. 35 (1982), 275--331.
• --. --. --. --., On the diffraction of waves by conical singularities, II, Comm. Pure Appl. Math. 35 (1982), 487--529.
• F. G. Friedlander, Sound Pulses, Cambridge Univ. Press, New York, 1958.
• F. G. Friedlander and R. B. Melrose, The wave front set of the solution of a simple initial-boundary value problem with glancing rays, II, Math. Proc. Cambridge Philos. Soc. 81 (1977), 97--120.
• P. GéRard and G. Lebeau, Diffusion d'une onde par un coin, J. Amer. Math. Soc. 6 (1993), 341--424.
• A. Hassell, R. Melrose, and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math. 181 (2004), 1--87.
• —, Scattering for symbolic potentials of order zero and microlocal propagation near radial points, preprint.v2[math.AP]
• L. HöRmander, The Analysis of Linear Partial Differential Operators, III, Grundlehren Math.Wiss. 274, Springer, Berlin, 1985.
• J. B. Keller, One hundred years of diffraction theory, IEEE Trans. Antennas and Propagation 33 (1985), 123--126.
• G. Lebeau, Propagation des ondes dans les variétés à coins'' Séminaire sur les équations aux dérivées partielles, 1995--1996., Sémtin. Éqn. Deriv. Partielles, École Polytech., Palaiseau, France, 1996, exp. no. XVI.
• —, Propagation des ondes dans les variétés à coins, Ann. Sci. École Norm. Sup. (4) 30 (1997), 429--497.
• R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1615--1664.
• R. B. Melrose, Microlocal parametrices for diffractive boundary value problems, Duke Math. J. 42 (1975), 605--635.
• —, Transformation of boundary problems, Acta Math. 147 (1981), 149--236.
• —, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces'' in Spectral and Scattering Theory (Sanda, 1992), Lecture Notes in Pure and Appl. Math. 161 Dekker, New York, 1994, 85--130.
• R. B. Melrose and J. SjöStrand, Singularities of boundary value problems, I, Comm. Pure Appl. Math. 31 (1978), 593--617.
• —, Singularities of boundary value problems, II, Comm. Pure Appl. Math. 35 (1982), 129--168.
• R. B. Melrose and J. Wunsch, Propagation of singularities for the wave equation on conic manifolds, Invent. Math. 156 (2004), 235--299.
• J. SjöStrand, Propagation of analytic singularities for second order Dirichlet problems, Comm. Partial Differential Equations 5 (1980), 41--93.
• —, Propagation of analytic singularities for second order Dirichlet problems, II, Comm. Partial Differential Equations 5 (1980), 187--207.
• —, Propagation of analytic singularities for second order Dirichlet problems, III, Comm. Partial Differential Equations 6 (1981), 499--567.
• A. Sommerfeld, Mathematische Theorie der Diffraktion, Math. Ann. 47 (1896), 317--374.
• M. E. Taylor, Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math. 29 (1976), 1--38.
• A. Vasy, Propagation of singularities for the wave equation on manifolds with corners, to appear in Ann. Math.