Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian

Jamel El Kamel[1]; Chokri Yacoub[1]

  • [1] Faculty of Science of Monastir Department of Mathematics Boulevard de l’environnement Monastir Tunisia

Annales mathématiques Blaise Pascal (2005)

  • Volume: 12, Issue: 1, page 147-160
  • ISSN: 1259-1734

Abstract

top
In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra c -function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.

How to cite

top

El Kamel, Jamel, and Yacoub, Chokri. "Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian." Annales mathématiques Blaise Pascal 12.1 (2005): 147-160. <http://eudml.org/doc/10508>.

@article{ElKamel2005,
abstract = {In this paper we consider the modified wave equation associated with a class of radial Laplacians $L$ generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra $\mathbf\{c\}$-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.},
affiliation = {Faculty of Science of Monastir Department of Mathematics Boulevard de l’environnement Monastir Tunisia; Faculty of Science of Monastir Department of Mathematics Boulevard de l’environnement Monastir Tunisia},
author = {El Kamel, Jamel, Yacoub, Chokri},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Laplace-Beltrami operator; Damek-Ricci spaces},
language = {eng},
month = {1},
number = {1},
pages = {147-160},
publisher = {Annales mathématiques Blaise Pascal},
title = {Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian},
url = {http://eudml.org/doc/10508},
volume = {12},
year = {2005},
}

TY - JOUR
AU - El Kamel, Jamel
AU - Yacoub, Chokri
TI - Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian
JO - Annales mathématiques Blaise Pascal
DA - 2005/1//
PB - Annales mathématiques Blaise Pascal
VL - 12
IS - 1
SP - 147
EP - 160
AB - In this paper we consider the modified wave equation associated with a class of radial Laplacians $L$ generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra $\mathbf{c}$-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.
LA - eng
KW - Laplace-Beltrami operator; Damek-Ricci spaces
UR - http://eudml.org/doc/10508
ER -

References

top
  1. J.Ph. Anker, E. Damek, Ch. Yacoub, Spherical analysis on harmonic AN groups, Ann. Scuola Norm. Sup. Pisa XXIII (1996), 643-679 Zbl0881.22008MR1469569
  2. W.R. Bloom, Z. Xu, Fourier transforms of Schwartz functions on Chébli-Triméche hypergroups, Mh. Math. 125 (1998), 89-109 Zbl0893.43001MR1604930
  3. T.P. Branson, G. Olafsson, H. Schlichtkrull, Huygens’ principle in Riemannian symmetric spaces, Math. Ann. 301 (1995), 445-462 Zbl0822.43002
  4. T.P. Branson, G. Olafsson, Equipartition of energy for waves in symmetric spaces, J. Funct. Anal. 97 (1991), 403-416 Zbl0796.43006MR1111189
  5. T.P. Branson, Eventual partition of conserved quantities in wave motion, J. Math. Anal. Appl. 96 (1983), 54-62 Zbl0591.35036MR717494
  6. H. Chébli, Théorème de Paley-Wiener associé à un opérateur singulier sur ( 0 , ∞ ) , J. Math. Pures Appl. 58 (1979), 1-19 Zbl0394.34019MR533233
  7. R.J. Duffin, Equipartition of energy in wave motion, J. Math. Anal. Appl. 32 (1970), 386-391 Zbl0223.35055MR269190
  8. S. Helgason, Huygens’ principle for wave equation on symmetric spaces, J. Funct. Anal. 107 (1992), 279-288 Zbl0757.58036
  9. T.H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions : Group theoretical aspects and applications (1984), 1-85, Askey & al.R.A.R. Zbl0584.43010MR774055
  10. P.D. Lax, R.S. Phillips, Scattering theory, (1967), Academic Press, New York Zbl0186.16301MR217440
  11. P.D. Lax, R.S. Phillips, Translation representations for the solution of the non-euclidean wave equation, Comm. Pure Appl. Math. 32 (1979), 617-667 Zbl0425.35065MR533296
  12. G. Olafsson, H. Schlichtkrull, Wave propagation on Riemannian symmetric spaces, J. Funct. Anal. 107 (1992), 270-278 Zbl0757.58037MR1172024
  13. K. Trimèche, Transformation intégrale de Weyl et thóréme de Paley-Wiener associés à un opérateur différentiel singulier sur ( 0 , + ∞ ) , J. Math. Pures. Appl. 60 (1981), 51-98 Zbl0416.44002MR616008
  14. K. Trimèche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comp. Harm. Anal. 4 (1997), 97-112 Zbl0872.34059MR1429682
  15. Ch. Yacoub, (1994) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.