Necessary conditions for strong hyperbolicity of first order systems
The author considers a first order differential operator \(L\) acting on \(C^ \infty(\Omega, \mathbb{C}^ m)\) with Cauchy data on a non- characteristic surface (where \(\Omega\) is a nonempty and open subset in \(\mathbb{R}^{n+1}\)) and gives some necessary conditions in order that \(L+B\) is correctly posed for each \(B\in C^ \infty (\Omega, M_ m(\mathbb{C}))\), i.e. that \(L\) is strongly hyperbolic. Namely, he proves that: If \(h\) and \(M=(m_{ij})\) are the determinant and the cofactor matrix of the principal symbol of \(L\) respectively, and \(L\) is strongly hyperbolic, then the Cauchy problem for \(h+m_{ij}\) is correctly posed for every \(m_{ij}\).
scientific article; zbMATH DE number 528674
scientific article