Propagation of singularities for the wave equation on manifolds with corners

Vasy, András¹ 

¹ Department of Mathematics, MIT and Northwestern University

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 20, 16 p.

Résumé

In this talk we describe the propagation of ${𝒞}^{\infty }$ and Sobolev singularities for the wave equation on ${𝒞}^{\infty }$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times {ℝ}_t$, $P=D_t^2-{\Delta }_M$, and $u\in H_{\text{loc}}^1\left(X\right)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that ${WF}_{\text{b}}\left(u\right)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${𝒞}^{\infty }$ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [7]. 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).

These notes are a summary of [17], where the detailed proofs appear.

Détail

Classification : 58J47, 35L20

Bibliographie

[1] J. Cheeger and M. Taylor. Diffraction by conical singularities, I, II. Comm. Pure Applied Math., 35 :275–331, 487–529, 1982. | Zbl

[2] J. J. Duistermaat and L. Hörmander. Fourier integral operators, II. Acta Mathematica, 128 :183–269, 1972. | Zbl | MR

[3] A. Hassell, R. B. Melrose, and A. Vasy. Scattering for symbolic potentials of order zero and microlocal propagation near radial points. Preprint, 2005.

[4] A. Hassell, R. B. Melrose, and A. Vasy. Spectral and scattering theory for symbolic potentials of order zero. In Séminaire : Équations aux Dérivées Partielles, 2000–2001, Sémin. Équ. Dériv. Partielles, pages Exp. No. XIII, 21. École Polytech., Palaiseau, 2001. | Zbl | MR | mathdoc-id

[5] L. Hörmander. Fourier integral operators, I. Acta Mathematica, 127 :79–183, 1971. | Zbl | MR

[6] L. Hörmander. The analysis of linear partial differential operators, vol. 1-4. Springer-Verlag, 1983.

[7] G. Lebeau. Propagation des ondes dans les variétés à coins. Ann. Scient. Éc. Norm. Sup., 30 :429–497, 1997. | Zbl | MR | mathdoc-id

[8] R. B. Melrose and J. Sjöstrand. Singularities of boundary value problems. I. Comm. Pure Appl. Math, 31 :593–617, 1978. | Zbl | MR

[9] R. B. Melrose and J. Sjöstrand. Singularities of boundary value problems. II. Comm. Pure Appl. Math, 35 :129–168, 1982. | Zbl | MR

[10] R. B. Melrose, A. Vasy, and J. Wunsch. Propagation of singularities for the wave equation on manifolds with edges. In preparation.

[11] R. B. Melrose and J. Wunsch. Propagation of singularities for the wave equation on conic manifolds. Invent. Math., 156(2) :235–299, 2004. | Zbl | MR

[12] R. B. Melrose. Transformation of boundary problems. Acta Math., 147(3-4) :149–236, 1981. | Zbl | MR

[13] R. B. Melrose and P. Piazza. Analytic $K$-theory on manifolds with corners. Adv. Math., 92(1) :1–26, 1992. | Zbl | MR

[14] M. Mitrea, M. Taylor, and A. Vasy. Lipschitz domains, domains with corners and the Hodge Laplacian. Preprint, 2004. | MR

[15] A. Vasy. Propagation of singularities in many-body scattering. Ann. Sci. École Norm. Sup. (4), 34 :313–402, 2001. | Zbl | MR | mathdoc-id

[16] A. Vasy. Propagation of singularities in many-body scattering in the presence of bound states. J. Func. Anal., 184 :177–272, 2001. | Zbl | MR

[17] A. Vasy. Propagation of singularities for the wave equation on manifolds with corners. Preprint, 2004.