Skip to main content
SpringerLink
Log in

Strictly Hyperbolic Operators and Approximate Energies

  • Chapter
Analysis and Applications — ISAAC 2001

Part of the book series: International Society for Analysis, Applications and Computation ((ISAA,volume 10))

Abstract

In this note we collect some results on Sobolev-, CO∞- and Gevrey-well-posedness of the Cauchy problem for linear strictly hyperbolic operators having non Lipschitz-continuous coefficients. These results are obtained modifying the classical method of the energy estimates by the introduction of the so-called approximate energies, i.e. a family of energies which depend on a small parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. d’Alembert, J.: Traité de dynamique. Paris, 1743.

    Google Scholar 

  2. d’Alembert, J.: Recherches sur la courbe que forme une corde tendue, mise en vibration. Hist. Ac. Sci. Berlin, 1747, 3 (1749), 214–219.

    Google Scholar 

  3. d’Alembert, J.: Suites des Recherches sur la courbe que forme une corde tendue, mise en vibration. Hist. Ac. Sci. Berlin, 1747, 3 (1749), 220–229.

    Google Scholar 

  4. d’Alembert, J.: Réflexions sur la cause générale des vents. Paris, 1747.

    Google Scholar 

  5. d’Alembert, J.: Addition au mémoire sur la courbe que forme une corde tendue, mise en vibration, Hist. Ac. Sci. Berlin, 1750, 6 (1752), 355–360.

    Google Scholar 

  6. d’Alembert, J.: Recherches sur les vibrations des cordes sonores. O-pustules mathématiques, I, Paris, 1761, 1–64.

    Google Scholar 

  7. Alinhac, S.: Blowup for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston 1995.

    Google Scholar 

  8. Bernoulli, D.: Réflexions et éclaircissemens sur les nouvelles vibrations des cordes. Hist. Ac. Sci. Berlin, 1753, 9 (1755), 147–172.

    Google Scholar 

  9. Bernoulli, D.: Sur le mélange de plusieurs espéces de vibrations simples isochrones, qui peuvent coexister dans un même système de corps. Hist. Ac. Sci. Berlin, 1753, 9 (1755), 173–195.

    Google Scholar 

  10. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  11. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. École Norm. Sup. (4) 14 (1981), 209–246.

    MathSciNet  MATH  Google Scholar 

  12. Bony, J.-M., Shapira, P.: Existence et prologement des solutions holomorphes des équations aux dérivées partielles. Invent. Math. 17 (1972), 95–105.

    Article  MathSciNet  MATH  Google Scholar 

  13. Bony, J.-M., Shapira, P.: Existence et prolongement des solutions analytiques des systèmes hyperboliques non stricts. C. R. Acad. Sci. Paris 274 (1972), 86–89.

    MATH  Google Scholar 

  14. Bony, J.-M., Shapira, P.: Problème de Cauchy, existence et prolongement pour les hyperfonctions solutions des équations hyperboliques non strictes. C. R. A-cad. Sci. Paris 274 (1972), 188–191.

    MATH  Google Scholar 

  15. Burgatti, P.: Sull’estensione del metodo d’integrazione di Riemann all’equazioni lineari d’ordine n con due variabili indipendenti. Rend. Reale Accad. Lincei (5) 15 (1906), 602–609.

    MATH  Google Scholar 

  16. Calderon, A.P., Zygmund, A.: Singular integral operators and differential equations. Amer. J. Math. 79 (1957), 901–921.

    Article  MathSciNet  MATH  Google Scholar 

  17. Cauchy, L.-A.: Mémoire sur l’intégration des équations linéaires aux différentielles partielles et à coefficients constants. Oeuvres Complètes, 2e série, Gauthier-Villars, Paris, 1905.

    Google Scholar 

  18. Cicognani, M.: The Cauchy problem for strictly hyperbolic operators with non-absolutely continuous coefficients. to appear in Tsukuba J. Math.

    Google Scholar 

  19. Cohen, P: The non-uniqueness of the Cauchy problem. O. N. R. Techn Report 93, Stanford 1960.

    Google Scholar 

  20. Colombini, F., De Giorgi, E., Spagnolo, S.: Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temp. Ann. Sc. Norm. Sup. Pisa, 6 (1979), 511–559.

    MATH  Google Scholar 

  21. Colombini, F., Del Santo, D.: An example of non-uniqueness for a hyperbolic equation with non-Lipschitz-continuous coefficients. To appear.

    Google Scholar 

  22. Colombini, F., Del Santo, D., Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients, to appear in Ann. Sc. Norm. Sup. Pisa.

    Google Scholar 

  23. Colombini, F., Del Santo, D., Reissig, M.: On the optimal regularity of coefficients in hyperbolic Cauchy problems. To appear.

    Google Scholar 

  24. Colombini, F., Jannelli, E., Spagnolo, S.: Nonuniqueness in hyperbolic Cauchy problems. Ann. of Math. 126 (1987), 495–524.

    Article  MathSciNet  MATH  Google Scholar 

  25. Colombini, F., Lerner, N.: Hyperbolic operators with non-Lipschitz coefficients. Duke Math. J. 77 (1995), 657–698.

    MathSciNet  MATH  Google Scholar 

  26. Colombini, F., Spagnolo, S.: Sur la convergence de solutions d’équations paraboliques. J. Math. Pures Appl. 56 (1977), 263–305.

    MathSciNet  MATH  Google Scholar 

  27. Colombini, F., Spagnolo, S.: On the convergence of solutions of hyperbolic equations. Comm. Partial Differential Equations 3 (1978), 77–103.

    Article  MathSciNet  MATH  Google Scholar 

  28. Colombini, F., Spagnolo, S.: Second order hyperbolic equations with coefficients real analytic in space variables and discontinuous in time. J. Analyse Math. 38 (1980), 1–33.

    MathSciNet  MATH  Google Scholar 

  29. Colombini, F., Spagnolo, S.: Hyperbolic equations with coefficients rapidly oscillating in time: a result of nonstability. J. Differ. Equations 52 (1984), 24–38.

    Article  MathSciNet  Google Scholar 

  30. Colombini, F., Spagnolo, S.: Some examples of hyperbolic equations without local solvability. Ann. scient. Éc. Norm. Sup. (4) 22 (1989), 109–125.

    MathSciNet  MATH  Google Scholar 

  31. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York, 1, 1953, 2, 1962.

    Google Scholar 

  32. Courant, R., Lax, P.D.: Remarks on Cauchy’s problem for hyperbolic partial differential equations with constant coefficients in several independent variables. Comm Pure Appl. Math. 8 (1955), 497–502.

    MathSciNet  MATH  Google Scholar 

  33. De Giorgi, E.: Un esempio di non unicità della soluzione del problema di Cauchy relativo ad una equazione differenziale lineare a derivate parziali di tipo parabolico. Rend. Mat. 14 (1955), 382–387.

    MATH  Google Scholar 

  34. De Giorgi, E., Spagnolo, S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Unione Mat. Ital. (4) 8 (1973), 391–411.

    Google Scholar 

  35. Demidov, S.S.: Création et développement de la théorie des équations différentielles aux dérivées partielles dans les travaux de J. d’Alembert. Rev. Histoire Sci. Appl. 35 (1982), 3–42.

    Article  MathSciNet  Google Scholar 

  36. Egorov, Yu.V., Shubin, M.A.: Linear Partial Differential Equations. Foundations of the Classical Theory. Encyclopaedia of Mathematical Science 30, Springer, Berlin, 1991.

    Google Scholar 

  37. Euler, L.: Additamentum ad dissertationem de infinitis curvis ejusdem generis. Comm. Ac. Sci. Petrop. 1734–35. 7 (1740), 184–200.

    Google Scholar 

  38. Euler, L.: De vibratione chordarum exercitatio Hist. Ac. Sci. Berlin, 1748, 4 (1750), 68–85 (= Id., Opera ornnia, Ser. II, vol. 10, 63–77 ).

    Google Scholar 

  39. Euler, L.: Remarques sur les mémoires précédents de M. Bernoulli. Hist. Ac. Sci. Berlin, 1753, 9 (1755), 196–222 (= Id., Opera ornnia, Ser. II, vol. 10, 232–254 ).

    Google Scholar 

  40. Euler, L.: De chordis vibrantibus disquisitio ulterior. Novi comm. Ac. Sci. Petrop., 1772, 17 (1773), 381–409 (= Id., Opera ornnia,Ser. II, vol. 11/1, 62–80).

    Google Scholar 

  41. Friedrichs, K.O.: Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392.

    MathSciNet  MATH  Google Scholar 

  42. Friedrichs, K.O., Lewy, H.: Über die Eindeutigkeit und das Abhängigkeitsgebiet der Lösungen beim Anfangswertproblem linearer hyperbolischer Differentialgleichungen. Mat. Ann. 98 (1928), 192–204.

    Article  MathSciNet  Google Scholar 

  43. Fourier, J.: Ouvres. 2 voll., Gauthier-Villars, Paris, 1888–1890.

    Google Scholar 

  44. Gârding, L.: Linear hyperbolic partial differential equations with constant coefficients. Acta Math. 85 (1950), 1–62.

    Article  Google Scholar 

  45. Gârding, L.: Solution directe du problème de Cauchy pour les équations hyperboliques. Colloques Internationaux du Centre Nat. de la Recherche Scient. 71 (1956), 71–90.

    Google Scholar 

  46. Gârding, L.: L’inegalité de Friedrichs et Lewy pour les équations hyperboliques linéaires d’ordre superieur. C. R. Acad. Sci. Paris 239 (1954), 849–850.

    MathSciNet  MATH  Google Scholar 

  47. Gelfand, I.M.: Some questions of analysis and differential equations. Uspehi Mat. Nauk 14 (3) 1959), 3–19; Amer. Math. Soc. Transi. 26 (2) (1963), 201–219.

    Google Scholar 

  48. Hadamard, J.: Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann et Cie, Paris 1932.

    Google Scholar 

  49. Helmholtz, H.: Theorie der Luftschwingungen in Röhren mit offenen Enden. J. Reine Angew. Math. 57 (1860), 1–72.

    Article  MATH  Google Scholar 

  50. Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Sechste Mitteilung, Gött. Nachr. (1910), 355–419.

    Google Scholar 

  51. Holmgren, E.: Om Cauchys Problem vid de lineära partiella differentialekvationerna of 2:dra ordningen. Arkiv f. Mat., Astr. och Fys. 2 (1905), 1–13.

    MATH  Google Scholar 

  52. Holmgren, E.: Sur les systèmes linéaires aux dérivées partielles du premier ordre à charactéristiques réelles et distinctes. Arkiv f. Mat., Astr. och Fys. 6 (1909), 1–10.

    Google Scholar 

  53. Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin 1963.

    Book  MATH  Google Scholar 

  54. Hörmander, L.: The Analysis of Linear Partial Differential Operators, I-IV. Springer, Berlin 1983–1985.

    Google Scholar 

  55. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Math. et Appl. 26, Springer, Berlin 1997.

    Google Scholar 

  56. Hurd, A.E., Sattinger, D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Amer. Math. Soc. 132 (1968), 159–174.

    Article  MathSciNet  MATH  Google Scholar 

  57. Kirchhoff, G.: Zur Theorie der Lichtstrahlen. Sitzungsber. Akad. Wiss. zu Berlin 1882, 641–669.

    Google Scholar 

  58. Ivrii, V.Ja.: Linear Hyperbolic Equations. Encyclopaedia of Mathematical Science, 33, Springer, Berlin 1993.

    Google Scholar 

  59. Jannelli, E.: Regularly hyperbolic systems and Gevrey classes. Ann. Mat. Pura. Appl. 140 (1985), 133–145.

    Article  MathSciNet  MATH  Google Scholar 

  60. John, F.: Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience, New York 1955.

    MATH  Google Scholar 

  61. John, F.: Hyperbolic and Parabolic Equations. L. Bers, F. John and M. Schechter, Partial differential equations. Proceedings of the Summer Seminar, Boulder, Colorado, 1957. Lectures in Applied Mathematics, III. Interscience Publishers John Wiley and Sons, Inc. New York-London-Sydney 1964.

    Google Scholar 

  62. Izdat. Nauka, Moscow 1968 ( Russian).

    Google Scholar 

  63. Kline, M.: Mathematical thought from ancient to modern times. Oxford University Press, New York 1972.

    Google Scholar 

  64. Kleinerman, S, Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Comm. Pure Appl. Math. 33 (1980), 241–263.

    Google Scholar 

  65. Kubo, A., Reissig, M.:Construction of parametrix for hyperbolic equations with slow oscillations in non-Lipschitz coefficients. To appear.

    Google Scholar 

  66. Kubo, A., Reissig, M.: C°°-well posedness of the Cauchy problem for quasilinear hyperbolic equations with coefficients non-Lipschitz in t and smooth in x. To appear.

    Google Scholar 

  67. Lagrange, J.-L.: Nouvelles recherches sur la nature et la propagation du son. Misc. Taur., 1760–1761, 2 (1762), 11–172 (= Id., OEuvres, t. I, Paris, 1867, 15 1316 ).

    Google Scholar 

  68. Lagrange, J.-L.: Misc. Taur., t. 13, 1759, i-x, 1–112 (= Id., OEuvres,t. I, Paris, 39–148).

    Google Scholar 

  69. Lax, P.D.: Asymptotics solutions of oscillatory initial value problems. Duke Math. J. 24 (1957), 627–646.

    MathSciNet  MATH  Google Scholar 

  70. Lax, P.D.: Differential equations, difference equations and matrix theory. Comm Pure Appt. Math. 11 (1958), 175–194.

    MathSciNet  MATH  Google Scholar 

  71. Lax, P.D.: Lectures on hyperbolic differential equations. Stanford University, 1963.

    Google Scholar 

  72. Leray, J.: Lectures on hyperbolic equations with variable coefficients. Inst. Advanced Study, Princeton, 1952.

    Google Scholar 

  73. Leray, J.: On linear hyperbolic differential equations with variable coefficients on a vector space. Ann. Math. Studies 33 (1954), 201–210.

    MathSciNet  MATH  Google Scholar 

  74. Leray, J.: Équations hyperboliques non-strictes: contre-exemples, du type De Giorgi, aux théorèmes d’existence et d’unicité. Mat. Ann. 162 (1966), 228–236.

    Article  MathSciNet  MATH  Google Scholar 

  75. Levi, E.E.: Sul problema di Cauchy. Rend. Reale Accad. Lincei (5) 16 (1907), 105–112.

    MATH  Google Scholar 

  76. Levi, E.E.: Sulle equazioni lineari totalmente ellittiche alle derivate parziali. Rend. Circ. Mat. Palermo 24 (1907), 275–317.

    Article  MATH  Google Scholar 

  77. Levi, E.E.: Sul problema di Cauchy per le equazioni lineari in due variabili a caratteristiche reali. Rend. R. Ist. Lomb. di Scienze e Lettere (2) 41 (1908), 408–428.

    MATH  Google Scholar 

  78. Magnus, W., Winkler, S.: Hill’s Equation. Wiley, New York 1966.

    Google Scholar 

  79. Martineau, A.: Sur les functionnelles analytiques et la transformation de Fourier-Borel. J. Analyse Math. 11 (1963), 1–164.

    Article  MathSciNet  MATH  Google Scholar 

  80. Mizohata, S.: Le problème de Cauchy pour les systèmes hyperboliques et paraboliques. Mem. Coll. Sci. Univ. Kyoto, Ser. A, 32 (1959), 181–212.

    MathSciNet  MATH  Google Scholar 

  81. Mizohata, S.: Systèmes hyperboliques. J. Math. Soc. Japan 11 (1959), 205–233.

    Article  MathSciNet  MATH  Google Scholar 

  82. Mizohata, S.: The theory of partial differential equations. University Press, Cambridge 1973.

    MATH  Google Scholar 

  83. Nicoletti, O.: Sulla estensione dei metodi di Picard e di Riemann ad una classe di equazioni a derivate parziali. Atti della R. Accademia delle scienze fisiche e mat. di Napoli (2) 8 (1897), 1–22.

    Google Scholar 

  84. Nishitani, T.: Sur les équations hyperboliques h coefficients höldériens en t et de classe de Gevrey en z. Bull. Sci. Math. (2) 107 (1983), 113–138.

    MathSciNet  MATH  Google Scholar 

  85. Petrowski, I.G.: Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen. Mat. Sbornik 2 (44) (1937), 815–866.

    Google Scholar 

  86. Petrowski, I.G.: Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen im Gebiete der nichtanalytischen Funktionen. Bull. Univ. Moscou, Sér. Int., Sect. A, 1 (7) (1938), 1–74.

    Google Scholar 

  87. Petrowski, I.G.: Lectures on partial differential equations. 2nd ed. Gos. Izd. Tekh. Teor. Lit., Moscow 1953. English translation, Interscience, New York 1954.

    Google Scholar 

  88. Picard, É.: Mémoires sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J. Math. Pures Appl. (4) 6 (1890), 145–210.

    Google Scholar 

  89. Plis, A.: The problem of uniqueness for the solutions of a system of partial differential equations. Bull. Acad. Pol. Sci. 2 (1954), 55–57.

    MathSciNet  MATH  Google Scholar 

  90. Poisson, D.: Mémoire sur la théorie du son. Journal de l’École Polytechn. 7 (1807), 319–392.

    Google Scholar 

  91. Poisson, D.: Mém. de l’Acad. des Sci., Paris, (2) 3 (1818), 121–176.

    Google Scholar 

  92. Rellich, F.: Verallgemeinerung der Riemannschen Integrationsmethode auf Differentialgleichungen n-ter Ordnung in zwei Veränderlichkeiten. Math. Ann. 103 (1930), 249–278.

    Article  MathSciNet  MATH  Google Scholar 

  93. Riemann, B.: Ober die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandl. Königl. Ges. Wiss. Göttingen 8 (1850).

    Google Scholar 

  94. Rubinowicz, A.: Herstellung von Lösungen gemischter Randwertprobleme bei hyperbolischen Differentialgleichungen zweiter Ordnung durch Zusammenstückelung aus Lösungen einfacherer gemischter Randwertaufgaben. Monatsh. Math. 30 (1920), 65–79.

    Article  MathSciNet  MATH  Google Scholar 

  95. Rubinowicz, A.: Eindeutigkeit der Lösungen der Maxwellschen Gleichungen. Phyzik. Z. 27 (1926), 707–710.

    Google Scholar 

  96. Schauder, J.: Das Anfangswertproblem einer quasilinearen hyperbolischen Differentialgleichung zweiter Ordnung. Fund. Math. 24 (1935), 213–246.

    Google Scholar 

  97. Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa, 22 (1968), 571–597.

    MATH  Google Scholar 

  98. Spagnolo, S.: Convergence in energy for elliptic operators. Proc. 3rd Symp. numer. Solut. partial Differ. Equat., College Park 1975, (1976), 469–498.

    MathSciNet  Google Scholar 

  99. Truesdell, C.: The rational mechanics of flexible or elastic bodies. 1638–1788, dans L. Euler, Opera omnia, Ser. II, 11, Turici, 1960.

    Google Scholar 

  100. Wallner, C.R.: Totale und partielle Differentialgleichungen. M. Cantor, Vorlesungen über Geschichte der Mathematik, Bd. 4, Leipzig, 1908, 871–1074.

    Google Scholar 

  101. Wieleitner, H.: Geschichte der Mathematik. II. Teil, Leipzig, 1911.

    Google Scholar 

  102. Zaremba, S.: Sopra un teorema d’unicità relativo alla equazione delle onde sferiche. Rend. Reale Accad. Lincei (5) 24 (1915), 904–908.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Via F. Buonarroti 2, 56127, Pisa, Italy

    Ferruccio Colombini

  2. Dipartimento di Scienze Matematiche, Università di Trieste, Via A. Valerio 12/1, 34127, Trieste, Italy

    Daniele Del Santo

Authors
  1. Ferruccio Colombini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniele Del Santo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Heinrich G. W. Begehr

  2. University of Delaware, Newark, DE, USA

    Robert P. Gilbert

  3. York University, Toronto, Canada

    Man Wah Wong

Additional information

Dedicated to Kunihiko Kajitani on the occasion of his sixtieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Colombini, F., Del Santo, D. (2003). Strictly Hyperbolic Operators and Approximate Energies. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_17

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3741-7_17

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5247-9

  • Online ISBN: 978-1-4757-3741-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation