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.”
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References
Boman, J., On the propagation of analyticity of solutions of differential equations with constant coefficients. Ark. för Matematik 5 (1964), 271–279.
Hörmander, L., On the singularities of solutions of partial differential equations with constant coefficients. Isral J. Math. 13 (1972 ), 82–105.
Zerner, M., Solutions de l’équation des ondes présentant des singularités sur une droite. C. R. Acad. Sci. Paris 250 (1960), 2980–2982.
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Hörmander, L. (1985). Commentary on [39] and [47]. In: Moser, J. (eds) Fritz John. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-5406-5_34
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DOI: https://doi.org/10.1007/978-1-4612-5406-5_34
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