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Fritz John pp 118-162 | Cite as

On Linear Partial Differential Equations with Analytic Coefficients

Unique Continuation of Data
  • Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

Abstract

In the theory of partial differential equations solutions are commonly determined, which satisfy additional conditions, e.g. assume given Cauchy data or boundary values on a certain surface. Of paramount importance from the point of view of both theory and applications is the question to what extent the solution is determined by the data, and how far the data can be described arbitrarily. It is well known that for different types of equations different sets of data appear to be appropriate. In this paper the question of unique determination of a solution by Cauchy data and the arbitrariness of the data will be treated for the most general linear equation with analytic coefficients. The theorems derived will be valid for any type or order of the equation and for any number of independent variables. Simple conditions on the equation and initial surface will be given, which are necessary if a solution of the Cauchy problem for arbitrary data is to exist. The impossibility to solve the Cauchy problem for certain types of equations will be recognized as tied up with the functional character of the solutions, which belong to a class of functions with the property of “unique continuation” similar to the analytic functions. Before describing the results of this paper we shall review some of the known properties of solutions and introduce the descriptive terminology that will be used here.

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Bibliography

  1. 1.
    D’Adhémar, R., Sur une classe d’equations aux derives partielles du second ordre du type hyperbolique, à 3 ou 4 variables indépendantes. Thesis, Paris, 1904.Google Scholar
  2. 2.
    Bernstein, S., Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre. Mathematische Annalen, Volume 59, pp. 20–76, 1904.CrossRefGoogle Scholar
  3. 3.
    Carleman, T., Sur un problèm d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Arkiv för Matematik, Astronomi och. Fysik, Volume 26 B, pp. 1–9, 1939.Google Scholar
  4. 4.
    Courant, R., and Hilbert, D., Methoden der mathematischen Physik. Volume II, Interscience (reprinted ), 1937.CrossRefGoogle Scholar
  5. 5.
    Friedrich, K. O., and Lewy, H., Über die Eindeutigkeit und das Abhägigkeitsgebiet der Lösungen beim Anfangswetproblem linearer hyperbolischer Differentialgleichungen. Mathematische Annalen, Volume 98, pp. 192–204, 1927.CrossRefGoogle Scholar
  6. 6.
    Hadamard, J., Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann et Cie, Paris, 1932.Google Scholar
  7. 7.
    Hadamard, J., Sur le cas anormal du problème de Cauchy pour l’équation des ondes. In Studies and Essays presented to R. Courant on his 60th birthday, Interscience, New York, pp. 161–165, 1948.Google Scholar
  8. 8.
    Herglotz, G., Über die Integration linearer, partieller Differentialgleichungen mit konstanten Koefizienten. Berichte der mathematischen-physischen Klassen der sächsischen Akademie, Volume 78, 1926, and Volume 80, 1928;Google Scholar
  9. 8.
    Abhandlungen aus dem mathematischen Seminar der hamburgischen Universität, Volume 6, pp. 189–197, 1928.Google Scholar
  10. 9.
    Holmgren, E., Über Systeme von linearen partiellen Differentialeichungen. Öfversigt af kongl. Vetenskaps-Akademiens Förhandlingar, Volume 58, pp. 91–103, 1901.Google Scholar
  11. 10.
    Hopf, E., Über den funktionalen, insbensondere den analytischen Charakter der Lösungen elliptischer Differentialgleichungen zweiter Ordnung. Mathematische Zeitschrift, Volume 34, pp. 194–233, 1931.CrossRefGoogle Scholar
  12. 11.
    John F., Abhängigkeiten zwischen den Flächenintegralen einer stetigen Funktion. Mathematische Annalen, Volume 111, pp. 541–559, 1935.CrossRefGoogle Scholar
  13. 12.
    John F., Linear partial differential equations with analytic coefficients. Proceedings of the National Academy of Sciences, Volume 29, pp. 94–104, 1943.CrossRefGoogle Scholar
  14. 13.
    Lewy, H., Eindeutigkeit der Lösung des Anfangswertproblems einer elliptischen Differentialgleichung zweiter Ordnung in zwei Veränderlichen. Mathematische Annalen, Volume 104, pp. 325–339, 1931.CrossRefGoogle Scholar
  15. 14.
    Lewy, H., Neuer Beweis des analytischen Charakters der Lösungen elliptischer Differentialgleichungen. Mathematische Annalen, Volume 101, pp. 609–619, 1929.CrossRefGoogle Scholar
  16. 15.
    Lewy, H., Sur une nouvelle formule dans les équations linéaires elliptiques et une application au problème de Cauchy. Comptes Rendus de l’Académie des Sciences pp. 112–113, 1933.Google Scholar
  17. 16.
    Owens, G., An explicit formula for the solution of the ultrahyperbolic equation in four independent variables. Duke Mathematical Journal, Volume 9, pp. 272–282, 1942.CrossRefGoogle Scholar
  18. 17.
    Owens, G., Uniqueness of solution of ultrahyperbolic partial differential equations. American Journal of Mathematics, Volume 69, pp. 184–188, 1947.CrossRefGoogle Scholar
  19. 18.
    Petrowsky, I., Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen im Gebiete der nichtanalytischen Funktionen. Bulletin de l’Université de Moscou, Série Internationale, Section A, Volume I, pp. 1–74, 1938.Google Scholar
  20. 19.
    Radó, T., Das Hilbertsche Theorem über den analytischen Charakter der Lösungen der partiellen Differentialgleichungen zweiter Ordnung. Mathematische Zeitschrift, Volume 25, pp. 514–589, 1926.CrossRefGoogle Scholar
  21. 20.
    Titt, E. W., An initial value problem for all hyperbolic partial differential equations of second order with three independent variables. Annals of Mathematics, Volume 40, pp. 862–891, 1939.CrossRefGoogle Scholar
  22. 21.
    Volterra, V., Sur les vibrations des corps élastiques isotropes. Acta Mathematica, Volume XV III, 1894.Google Scholar

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© Springer Science+Business Media New York 1985

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  • Fritz John

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