Fritz John pp 335-352

Algebraic Conditions for Hyperbolicity of Systems of Partial Differential Equations

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

Abstract

Partial differential equations or systems of such equations are classified according to “type”, such as “elliptic”, “parabolic”, “hyperbolic”, etc. Hyperbolic systems (with respect to the initial plane t = 0) are those for which the initial value problem is well posed in the sense of Hadamard (see ). General conditions that are both necessary and sufficient for hyperbolicity are complicated (see , , , , , , , ). The situation, however, is rather simple for an m-th order linear homogeneous system of equations with constant coefficients:
$$P\left( {\frac{\partial }{{\partial t}},\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}} \right)u = 0$$
(1)
for a vector u = u(t, x 1 • • •, x n ) = u(t, x) with N components. Here P is an N x N square matrix whose elements p ik are m-th degree forms in their n + 1 arguments. We associate with (1) the matrix (the “symbol” of (1))
$$P(\lambda ,\xi ) = P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = (Pik(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}))$$
(2)
and the characteristic form
$$Q(\lambda ,\xi ) = Q(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n}) = \det P(\lambda ,{\xi _1} \cdot \cdot \cdot ,{\xi _n})$$
(3)
of degree mN in its arguments.

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