Fritz John pp 353-359 | Cite as

Addendum to: Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations

  • Fritz John
Part of the Contemporary Mathematicians book series (CM)


In the paper [18]† we dealt with a certain type of homogeneous second-order systems of partial differential equations (1) with constant coefficients. Here the matrix form P = P(λ, ξ) was represented by a point in ℝ81. We studied in particular the forms P in the set K ɛH, that is those P corresponding to a hyperbolic system (1) which lie in an ε-neighborhood of the special form P°. It was found that these P are never strictly hyperbolic and cannot even be approximated by strictly hyperbolic ones. We shall derive now a further property of the PK ɛH for ε sufficiently small, namely, that the corresponding differential equations (1) stay hyperbolic if we add any low-order terms with complex variable coefficients. This means that for systems with constant principal part P near P° the notions of weak and strong hyperbolicity coincide.6 The proof makes essential use of the fact proved earlier that the second derivatives of the function D*(ξ) for ξh near a singular point form a positive definite matrix.


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  1. [18]
    John, F., Algebraic conditions for hyperbolicity of systems of partial differential equations, Comm. Pure Appl. Math. 31, 1978, pp. 89–106.CrossRefGoogle Scholar
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    Yamaguti, M., and Kasahara, K., Sur le système hyperbolique à coefficients constants, Proc. Japan Acad. 35, 1959, pp. 547–550.CrossRefGoogle Scholar
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    Strang, G., On strong hyperbolicity, J. Math. Kyoto Univ. 6 (3), 1967, pp. 397–417.Google Scholar
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    Oleinik, O. A., On the Cauchy problem for weakly hyperbolic equations, Commun. Pure Appl. Math. 23, 1970, pp. 569–586.Google Scholar
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    Hörmander, L., The Cauchy problem for differential equations with double characteristics, J. Anal. Math. (Paris) 32, 1977, pp. 118–196.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 1985

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  • Fritz John

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