×

On lars Hörmander’s remark on the characteristic Cauchy problem. (English) Zbl 1124.35037

The author re-considers a result of L. Hörmander [J. Funct. Anal. 93, No. 2, 270–277 (1990; Zbl 0724.35060)], concerning a characteristic Cauchy problem for a class of wave equations on spatially compact space-times, with initial data on hypersurfaces that were weakly spacelike. The metric on the space-time and the first-order perturbation were assumed to be smooth. Here the author extends the result to a Lipschitz metric \(g^{jk}\), namely for the equation \(\partial^2_t u- g^{jk}\partial_j\partial_k u= 0\), with Einstein notation.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L05 Wave equation

Citations:

Zbl 0724.35060
PDF BibTeX XML Cite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Baez, J. C; Segal, I. E.; Zhou, Z. F., The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Anal., 93, 239-269, (1990) · Zbl 0724.35105
[2] Christodoulou, D; Klainerman, S., The global nonlinear stability of the Minkowski space, Princeton Mathematical, 41, x+514 pp., (1993) · Zbl 0827.53055
[3] Chrusciel, P.; Delay, E. E., Existence of non trivial, asymptotically vacuum, asymptotically simple space-times, Class. Quantum Grav., 19, L71-L79, (2002) · Zbl 1005.83009
[4] Chrusciel, P.; Delay, E. E., On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, 94, (2003), Mémoires de la S.M.F. · Zbl 1058.83007
[5] Corvino, J., Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., 214, 137-189, (2000) · Zbl 1031.53064
[6] Corvino, J.; Schoen, R. M., On the asymptotics for the vacuum Einstein constraint equations · Zbl 1122.58016
[7] Friedlander, F. G., Radiation fields and hyperbolic scattering theory, Math. Proc. Camb. Phil. Soc., 88, 483-515, (1980) · Zbl 0465.35068
[8] Friedlander, F. G., Notes on the wave equation on asymptotically Euclidean manifolds, J. Functional Anal., 184, 1-18, (2001) · Zbl 0997.58013
[9] Hörmander, L., A remark on the characteristic Cauchy problem, J. Funct. Anal., 93, 270-277, (1990) · Zbl 0724.35060
[10] Klainerman, S.; Nicolò, F., Peeling properties of asymptotically flat solutions to the Einstein vacuum equations, Class. Quantum Grav., 20, 14, 3215-3257, (2003) · Zbl 1045.83016
[11] Mason, L. J.; Nicolas, J.-P., Conformal scattering and the Goursat problem, J. Hyperbolic. Diff. Eq., 1, 2, 197-233, (2004) · Zbl 1074.83019
[12] Penrose, R., Null hypersurface initial data for classical fields of arbitrary spin and for general relativity, in Aerospace Research Laboratories report 63-56 (P.G. Bergmann), Vol. 12, (1963) · Zbl 0452.53014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.