On Lars Hörmander’s remark on the characteristic Cauchy problem

Jean-Philippe Nicolas[1]

  • [1] Université Bordeaux 1 Institut de Mathématiques, M.A.B. 351 cours de la Libération 33405 Talence Cedex (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 517-543
  • ISSN: 0373-0956

Abstract

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We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are L loc ∞ , with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential : essentially, a 𝒞 1 metric and a potential with continuous coefficients of the first order terms and locally L ∞ coefficients for the terms of order 0 .

How to cite

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Nicolas, Jean-Philippe. "On Lars Hörmander’s remark on the characteristic Cauchy problem." Annales de l’institut Fourier 56.3 (2006): 517-543. <http://eudml.org/doc/10156>.

@article{Nicolas2006,
abstract = {We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are $L^\infty _\mathrm\{loc\}$, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential : essentially, a $\{\mathcal\{C\}\}^1$ metric and a potential with continuous coefficients of the first order terms and locally $L^\infty $ coefficients for the terms of order $0$.},
affiliation = {Université Bordeaux 1 Institut de Mathématiques, M.A.B. 351 cours de la Libération 33405 Talence Cedex (France)},
author = {Nicolas, Jean-Philippe},
journal = {Annales de l’institut Fourier},
keywords = {Wave equation; Cauchy problem; characteristic Cauchy problem; very weak regularity},
language = {eng},
number = {3},
pages = {517-543},
publisher = {Association des Annales de l’institut Fourier},
title = {On Lars Hörmander’s remark on the characteristic Cauchy problem},
url = {http://eudml.org/doc/10156},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Nicolas, Jean-Philippe
TI - On Lars Hörmander’s remark on the characteristic Cauchy problem
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 517
EP - 543
AB - We extend the results of a work by L. Hörmander [9] concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are $L^\infty _\mathrm{loc}$, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential : essentially, a ${\mathcal{C}}^1$ metric and a potential with continuous coefficients of the first order terms and locally $L^\infty $ coefficients for the terms of order $0$.
LA - eng
KW - Wave equation; Cauchy problem; characteristic Cauchy problem; very weak regularity
UR - http://eudml.org/doc/10156
ER -

References

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  8. F. G. Friedlander, Notes on the wave equation on asymptotically Euclidean manifolds, J. Functional Anal. 184 (2001), 1-18 Zbl0997.58013MR1846782
  9. L. Hörmander, A remark on the characteristic Cauchy problem, J. Funct. Anal. 93 (1990), 270-277 Zbl0724.35060MR1073287
  10. S. Klainerman, F. Nicolò, Peeling properties of asymptotically flat solutions to the Einstein vacuum equations, Class. Quantum Grav. 20 (2003), 3215-3257 Zbl1045.83016MR1992002
  11. L. J. Mason, J.-P. Nicolas, Conformal Scattering and the Goursat problem, J. Hyperbolic. Diff. Eq. 1 (2004), 197-233 Zbl1074.83019MR2070126
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