Non-Lipschitz coefficients for strictly hyperbolic equations

Fumihiko Hirosawa[1]; Michael Reissig[2]

  • [1] Department of Mathematics Nippon Institute of Technology Saitama 345-8501, Japan
  • [2] Fakultät für Mathematik und Informatik TU Bergakademie Freiberg 09596 Freiberg, Germany

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 3, page 589-608
  • ISSN: 0391-173X

Abstract

top
In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get C ∞ well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.

How to cite

top

Hirosawa, Fumihiko, and Reissig, Michael. "Non-Lipschitz coefficients for strictly hyperbolic equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 589-608. <http://eudml.org/doc/84541>.

@article{Hirosawa2004,
abstract = {In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get $C^\infty $ well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.},
affiliation = {Department of Mathematics Nippon Institute of Technology Saitama 345-8501, Japan; Fakultät für Mathematik und Informatik TU Bergakademie Freiberg 09596 Freiberg, Germany},
author = {Hirosawa, Fumihiko, Reissig, Michael},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {589-608},
publisher = {Scuola Normale Superiore, Pisa},
title = {Non-Lipschitz coefficients for strictly hyperbolic equations},
url = {http://eudml.org/doc/84541},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Hirosawa, Fumihiko
AU - Reissig, Michael
TI - Non-Lipschitz coefficients for strictly hyperbolic equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 589
EP - 608
AB - In the present paper we explain the classification of oscillations and its relation to the loss of derivatives for a homogeneous hyperbolic operator of second order. In this way we answer the open question if the assumptions to get $C^\infty $ well posedness for weakly hyperbolic Cauchy problems or for strictly hyperbolic Cauchy problems with non-Lipschitz coefficients are optimal.
LA - eng
UR - http://eudml.org/doc/84541
ER -

References

top
  1. [1] R. Agliardi – M. Cicognani, Operators of p -evolution with non regular coefficients in the time variable, J. Differential Equations 202 (2004), 143-157. Zbl1065.35094MR2060535
  2. [2] M. Cicognani, Coefficients with unbounded derivatives in hyperbolic equations, to appear in Math. Nachr. Zbl1060.35071MR2100045
  3. [3] F. Colombini – E. De Giorgi – S. Spagnolo, Sur les equations hyperboliques avec des coefficients qui ne dependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 511-559. Zbl0417.35049MR553796
  4. [4] F. Colombini – D. Del Santo – T. Kinoshita, Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002), 327-358. Zbl1098.35094MR1991143
  5. [5] F. Colombini – D. Del Santo – M. Reissig, On the optimal regularity of coefficients in hyperbolic Cauchy problems, Bull. Sci. Math. 127 (2003), 328-347. Zbl1037.35038MR1988632
  6. [6] F. Colombini – N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657-698. Zbl0840.35067MR1324638
  7. [7] F. Hirosawa, On the Cauchy problem for second order strictly hyperbolic equations with non-regular coefficients, Math. Nachr. 256 (2003), 29-47. Zbl1026.35063MR1989376
  8. [8] F. Hirosawa – M. Reissig, Well-posedness in Sobolev spaces for second order strictly hyperbolic equations with non-differentiable oscillating coefficients, Ann. Global Anal. Geom. 25 (2004), 99-119. Zbl1047.35079MR2046768
  9. [9] S. Mizohata, “Theory of the partial differential equations”, Cambridge University Press, 1973. Zbl0263.35001MR599580
  10. [10] M. Reissig, Hyperbolic equations with non-Lipschitz coefficients, Rend. Sem. Mat. Univ. Politec. Torino 61 (2003), 135-182. Zbl1182.35155MR2056996
  11. [11] M. Reissig – K. Yagdjian, L p - L q estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients, Math. Nachr. 214 (2000), 71-104. Zbl1006.35057MR1762053
  12. [12] K. Yagdjian, “The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics, Micro-Local Approach”, Akademie-Verlag, Berlin, 1997. Zbl0887.35002MR1469977
  13. [13] T. Yamazaki, On the L 2 ( ℝ n ) well-posedness of some singular or degenerate partial differential equations of hyperbolic type, Comm. Partial Differential Equations 15 (1990), 1029-1078. Zbl0714.35053MR1070237

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.