Addendum to: Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations
In the paper † 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 ɛ ∩H, 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 P∈K ɛ∩H 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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