A sufficient condition for hyperbolicity of partial differential operators with constant coefficient principal part

Abstract

Let P P be a differential operator with principal part P m {P_m} , and suppose that P m {P_m} has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of P P when P P has constant coefficients, namely, that P P is weaker than P m {P_m} is also a sufficient condition for hyperbolicity in the case where P P does not have constant coefficients. Some generalizations are also made to the case where P P is a square matrix of differential operators.