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Fritz John pp 460-461 | Cite as

Commentary on [39] and [47]

  • L. Hörmander
Part of the Contemporary Mathematicians book series (CM)

Abstract

On page 553 in [47] John remarks that “there appears to be a connection between well-behaved character of a problem and regularity of the solution for regular data”. An example of this is given for the wave equation. u tt - u xx - u yy = 0 in R 3. On one hand, the continuation of solutions from a cylinder Φ = {(x, y, t); x 2 + y 2< r 2} to the complement is shown to be only very weakly continuous. On the other hand, a solution is constructed which is analytic in Φ, of class C m exactly on ∂Φ and of class C m +1 exactly in the complement of Φ. John comments on page 574: “What is remarkable is that this cylinder is not a characteristic surface for the differential equation. Apparently not all types of singularities propagate along characteristic surfaces.”

References

  1. [C. 1]
    Boman, J., On the propagation of analyticity of solutions of differential equations with constant coefficients. Ark. för Matematik 5 (1964), 271–279.CrossRefGoogle Scholar
  2. [C. 2]
    Hörmander, L., On the singularities of solutions of partial differential equations with constant coefficients. Isral J. Math. 13 (1972 ), 82–105.CrossRefGoogle Scholar
  3. [C. 3]
    Zerner, M., Solutions de l’équation des ondes présentant des singularités sur une droite. C. R. Acad. Sci. Paris 250 (1960), 2980–2982.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

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  • L. Hörmander

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