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Systems with Nondegenerate Characteristics

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2097)

Abstract

In this chapter we introduce the notion of nondegenerate multiple characteristics. Simple characteristics are nondegenerate characteristics of order 1. A double characteristic ρ of L is nondegenerate if and only if the rank of the Hessian at ρ of the determinant of L(x, ξ) is maximal. We prove that every hyperbolic system which is close to a hyperbolic system with nondegenerate multiple characteristic has a nondegenerate characteristic of the same order nearby. This implies that hyperbolic systems with a nondegenerate multiple characteristic can not be approximated by strictly hyperbolic systems which contrasts with the case of scalar hyperbolic operators. We also prove that if every multiple characteristic of the system L is nondegenerate then there exists a smooth symmetrizer and hence the Cauchy problem for L is C well posed for any lower order term. Finally we discuss about the stability of symmetric systems in the space of hyperbolic systems.

Keywords

  • Cauchy Problem
  • Constant Coefficient
  • Hyperbolic System
  • Diagonal Entry
  • Order System

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    We owe the proof of this lemma to T. Ibukiyama.

  2. 2.

    Another proof is found in [53].

  3. 3.

    The author owes this simple proof to A. Gyoja.

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Nishitani, T. (2014). Systems with Nondegenerate Characteristics. In: Hyperbolic Systems with Analytic Coefficients. Lecture Notes in Mathematics, vol 2097. Springer, Cham. https://doi.org/10.1007/978-3-319-02273-4_4

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