Skip to main content
SpringerLink
Log in

The Mixed Problem for Hyperbolic Systems

  • Conference paper
Singularities in Boundary Value Problems

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 65))

  • 193 Accesses

Abstract

Lax [27] and Mizohata [34] proved that for the non-characteristic Cauchy problem to be C∞ well-posed it is necessary that the characteristic roots are real. In the mixed problem Kajitani [23] obtained the results corresponding to those in the Cauchy problem under some restrictive assumptions. In §3 we shall relax his assumptions (see, also, [52]). We note that well-posedness of the mixed problems has been investigated by many authors ([1], [2], [19], [20], [22], [26], [32], [33], [38]). We shall consider the mixed problem for hyperbolic systems with constant coefficients in a quarter-space in §§4–6. Hersh [13], [l4], [15] studied the mixed problem for hyperbolic systems with constant coefficients. He gave the necessary and sufficient condition for the mixed problem to be C∞ well-posed. However, his proof seems to be incomplete (see [25]). Sakamoto [37] justified his results for single higher order hyperbolic equations (see, also, [40], [41] [42]). In §4 we shall consider C∞ well-posedness of the mixed problem. Duff [11] studied the location and structures of singularities of the fundamental solutions of the mixed problems for single higher order hyperbolic equations, using the stationary phase method.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
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. Agemi, R., and Shirota, T.: On necessary and sufficient conditions for L2-well-posedness of mixed problems for hyperbolic equations, J. Fac. Sci. Hokkaido Univ., 21 (1970), pp 133–151.

    MathSciNet  MATH  Google Scholar 

  2. Agemi, R., and Shirota, T: On necessary and sufficient conditions for L2-well-posedness of mixed problems for hyperbolic equations, II, J. Fac. Sci. Hokkaido Univ., 22 (1972), pp 137–149.

    MathSciNet  MATH  Google Scholar 

  3. Agmon, S., Douglis, A., and Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17 (1964), pp 3–92.

    Article  MathSciNet  Google Scholar 

  4. Andersson, K. G., and Melrose, R. B.: The propagation of singularities along gliding rays, Invent. Math., 41 (1977), pp 197–232.

    Article  MathSciNet  MATH  Google Scholar 

  5. Atiyah, M. F., Bott, R. and Gårding, L.: Lacunas for hyperbolic differential operators with constant coefficients, I, Acta Math., 12k (1970), pp 109–189.

    Article  Google Scholar 

  6. Beals, R.: Mixed boundary value problem for nonstrict hyperbolic equations, Bull. Amer. Math. Soc., 78 (1972), pp 520–521.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierstone, E.: Extension of Whitney fields from subanalytic set, Invent. Math., 46 (1978), pp 277–300.

    Article  MathSciNet  MATH  Google Scholar 

  8. Chazarain, J.: Problèmes de Cauchy abstraits et application à quelques problèmes mixtes, J. Functional Anal., 7 (l97l), pp 386–446.

    Article  MathSciNet  MATH  Google Scholar 

  9. Chazarain, J.: Construction de la paramétrix du problème mixte hyperbolique pour l’équation des ondes, C. R. Acad. Sci. Paris, 276 (1973), pp 1213–1215.

    MathSciNet  MATH  Google Scholar 

  10. Chazarain, J.: Reflection of C singularities for a class of operators with multiple characteristics, Publ. RIMS, Kyoto Univ., 12 Suppl. (1977), pp 39–52.

    Article  MathSciNet  Google Scholar 

  11. Duff, G. F. D.: On wave fronts, and boundary waves, Comm. Pure Appl, Math., 17 (1964), pp 189–225.

    Article  MathSciNet  MATH  Google Scholar 

  12. Eskin, G.: A parametrix for interior mixed problems for strictly hyperbolic equations, J. Anal. Math., 32 (1977), pp 17–62.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hersh, R.: Mixed problem in several variables, J. Math. Mech., 12 (1963), pp 317–334.

    MathSciNet  MATH  Google Scholar 

  14. Hersh, R.: Boundary conditions for equations of evolution, Arch. Rational Mech. Anal., 16 (1964)9 pp 243–264.

    MathSciNet  Google Scholar 

  15. Hersh, R.: On surface waves with finite and infinite speed of propagation, Arch. Rational Mech. Anal., 19 (1965), pp 308–316.

    Article  MathSciNet  MATH  Google Scholar 

  16. Hörmander, L.: “Linear Partial Differential Operators,” Springer-Verlag, Berlin, Gôttingen, Heidelberg, 1963.

    MATH  Google Scholar 

  17. Hörmander, L.: “On the singularities of solutions of partial differential equations,” International Conference of Functional Analysis and Related Topics, Tokyo, 1969.

    Google Scholar 

  18. Hörmander, L.: Uniqueness theorems and wave front set for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., 24 (1971), pp 671–703.

    Article  MathSciNet  MATH  Google Scholar 

  19. Ikawa, M.: Mixed problem for the wave equation with an oblique derivative boundary condition, Osaka J. Math., 7 (1970), pp 495–525.

    MathSciNet  MATH  Google Scholar 

  20. Ikawa, M.: Sur les problèmes mixtes pour l’équation des ondes, Publ. RIMS, Kyoto Univ., 10 (1975), pp 669–690.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ivrii, V. Ja., and Petkov, V. M.: Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed, Uspehi Mat. Nauk, 29 (1974), pp. 3–70. (Russian; English translation in Russian Math. Surveys.)

    Google Scholar 

  22. Kajitani, K.: Sur la condition nécessaire du problème mixte bien pose pour les systèmes hyperboliques à coefficients variables, Publ. RIMS, Kyoto Univ., 9 (1974), pp 261–284

    Article  MathSciNet  MATH  Google Scholar 

  23. Kajitani, K.: A necessary condition for the well posed hyperbolic mixed problem with variable coefficients, J. Math. Kyoto Univ., 14 (1974), pp 231–242.

    MathSciNet  MATH  Google Scholar 

  24. Kajitani, K.: On the E-well posed evolution equations, Comm. in Partial Differential Equations, 4 (1979), pp 595–608.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kasahara, K.: On weak well posedness of mixed problem for hyperbolic systems, Publ. RIMS, Kyoto Univ., 6 (1971), pp 503–514.

    Article  MathSciNet  MATH  Google Scholar 

  26. Kreiss, H.-O.: Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), pp 277–298.

    Article  MathSciNet  Google Scholar 

  27. Lax, P. D.: Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), PP 624–646.

    Article  MathSciNet  Google Scholar 

  28. Lejeune-Rifaut, E.: “Solutions” lémentaires des problèmes aux limites pour des opérateurs matriciels de dérivation hyperbolique posés dans un demi-espace,11 Dissertation, Liège, 1978.

    Google Scholar 

  29. Matsumura, M.: Localization theorem in hyperbolic mixed problems, Proc. Japan Acad., 47 (l97l), pp 115–119.

    Article  MathSciNet  MATH  Google Scholar 

  30. Melrose, R. B.: Microlocal parametrices for diffractive boundary value problems, Duke Math. J., 42 (1975), pp 605–635.

    Article  MathSciNet  MATH  Google Scholar 

  31. Melrose, R. B., and Sjöstrand, J.: Singularities of boundary problems I, Comm. Pure Appl. Math., 31 (1978), pp 593–617

    Article  MathSciNet  MATH  Google Scholar 

  32. Miyatake, S.: Mixed problem for hyperbolic equation of second order, J. Math. Kyoto Univ., 13 (1973), pp 435–487.

    MathSciNet  MATH  Google Scholar 

  33. Miyatake, S.: Mixed problems for hyperbolic equations of second order with first order complex boundary operators, Japanese J. Math., 1 (1975), pp 111–158.

    MathSciNet  Google Scholar 

  34. Mizohata, S.: Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961), pp 109–127.

    MathSciNet  MATH  Google Scholar 

  35. Mizohata, S.: On evolution equations with finite propagation speed, Israel J. Math., 13 (1972), pp 173–187.

    Article  MathSciNet  Google Scholar 

  36. Nirenberg, L.: “Lectures on Linear Differential Equations,” Regional Conference Series in Math., 20 (1973), Amer. Math. Soc., Providence, Rhode Island.

    Google Scholar 

  37. Sakamoto, R.: E-well posedness for hyperbolic mixed problems with constant coefficients, J. Math. Kyoto Univ. 14 (1974), pp 93–118.

    MathSciNet  MATH  Google Scholar 

  38. Sakamoto, R.: L2 -well-posedness for hyperbolic mixed problems, Publ. RIMS, Kyoto Univ., 8 (1972/73), pp 265–293.

    Google Scholar 

  39. Seeley, R. T.: Extension of C functions defined in a half space, Proc. Amer. Math. Soc., 15 (1964), pp 625–626.

    MathSciNet  MATH  Google Scholar 

  40. Shibata, Y.: A characterization of the hyperbolic mixed problems in a quarter space for differential operators with constant coefficients, Publ. RIMS, Kyoto Univ., 15 (1979), pp 357–399.

    Article  MATH  Google Scholar 

  41. Shibata, Y.: NOPROPORTIONAL E-well posedness of mixed initial-boundary value problem with constant coefficients in a quarter space, to appear.

    Google Scholar 

  42. Shirota, T.: On the propagation speed of hyperbolic operator with mixed boundary conditions, J. Fac. Sci. Hokkaido Univ., 22 (1972), pp 25–31.

    MathSciNet  MATH  Google Scholar 

  43. Sjöstrand, J.: Propagation of analytic singularities for second order Dirichlet problems, Comm. in Partial Differential Equations, 5 (1980), pp 41–94.

    Article  MATH  Google Scholar 

  44. Solonnikov, V. A.: On general boundary problems for systems which are elliptic in the sense of A. Douglis and L. Mrenberg. I, Izv. Akad. Nauk, 28 (1964), pp 665–706. (Russian; English translation in Amer. Math. Soc. Transi., 56 (1966).)

    Google Scholar 

  45. Taylor, M. S.: Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math., 29 (1976), pp 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  46. Tsuji, M.: Fundamental solutions of hyperbolic mixed problems with constant coefficients, Proc. Japan Acad., 51 (1975), pp 369–373.

    Article  MathSciNet  MATH  Google Scholar 

  47. Wakabayashi, S.: Singularities of the Riemann functions of hyperbolic mixed problems in a quarter-space, Proc, Japan Acad., 50 (1974), pp 821–825.

    Article  MathSciNet  MATH  Google Scholar 

  48. Wakabayashi, S.: Singularities of the Riemann functions of hyperbolic mixed problems in a quarter-space, Publ. RIMS, Kyoto Univ., 11 (1976), pp 417–440.

    Article  MathSciNet  MATH  Google Scholar 

  49. Wakabayashi, S.: Analytic wave front sets of the Riemann functions of hyperbolic mixed problems in a quarter-space, Publ. RIMS, Kyoto Univ., 11 (1976), pp 785–807.

    Article  MathSciNet  MATH  Google Scholar 

  50. Wakabayashi, S.: Microlocal parametrices for hyperbolic mixed problems in the case where boundary waves appear, Publ. RIMS, Kyoto Univ., 14 (1978), pp 283–307.

    Article  MathSciNet  MATH  Google Scholar 

  51. Wakabayashi, S.: Propagation of singularities of the fundamental solutions of hyperbolic mixed, problems, Publ. RIMS, Kyoto Univ., 15 (1979), pp 653–678.

    Article  MathSciNet  MATH  Google Scholar 

  52. Wakabayashi, S.: A necessary condition for the mixed problem to be C∞ well-posed, to appear in Comm. in Partial Differential Equations.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, the University of Tsukuba, Ibaraki, Japan

    Seiichiro Wakabayashi

Authors
  1. Seiichiro Wakabayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Inst. de Math., University of Liège, 15, avenue des Tilleuls, B-4000, Liege, Belgium

    H. G. Garnir

Rights and permissions

Reprints and permissions

Copyright information

© 1981 D. Reidel Publishing Company

About this paper

Cite this paper

Wakabayashi, S. (1981). The Mixed Problem for Hyperbolic Systems. In: Garnir, H.G. (eds) Singularities in Boundary Value Problems. NATO Advanced Study Institutes Series, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8434-9_14

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-8434-9_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-8436-3

  • Online ISBN: 978-94-009-8434-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation