Fritz John pp 353-359 | Cite as

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

## Abstract

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 ℝ_{81}. 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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## Bibliography

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