Hyperbolicity of localizations
Let \(P(x,D)\) be a differential operator of order \(m\) in an open set \(\Omega \subset \mathbb{R}^{n + 1}\) and hence a sum of differential polynomials \(P_ j (x,D)\) homogeneous of order \(j \leq m\) with symbols \(P_ j (x,\xi)\). Let \(z = (x,\xi)\) and \(z_ 0\) be a multiple characteristic point of the principal symbol \(P_ m (z)\). It is shown that if the Cauchy problem for \(P(x,D)\) is well-posed then it is possible to define the localization at \(z_ 0\) of \(P\) and it is proved that \(P_{z_ 0} (z)\) is a hyperbolic polynomial. Due to this fact one can iterate the argument, i.e. one can define the higher localizations \(P_{(z_ 0, z_ 1, \dots, z_ s)}\) as the localization of \(P_{(z_ 0, z_ 1, \dots, z_{s-1})}\) at the multiple characteristic point \(z_ s\). If the linearity of \(P_{m(z_ 0, z_ 1, \dots, z_ s)}\) is an involutive submanifold of the cotangent bundle of \(\Omega\) then it is proved that if the Cauchy problem for \(P(x,D)\) is correctly posed necessarily we must have that the higher localizations of \(P\) must coincide with the higher localizations of \(P_ m\). This result is very helpful in the case of operators exhibiting characteristics of higher multiplicity, in which very few is known about both necessary and sufficient conditions on the lower order terms (Levi conditions) related to the well-posedness of the Cauchy problem.
scientific article; zbMATH DE number 556418