Construction of Parametrix to Strictly Hyperbolic Cauchy Problems with Fast Oscillations in NonLipschitz Coefficients†

Abstract

The goal of this article is to construct parametrix to strictly hyperbolic Cauchy problems with nonLipschitz coefficients depending on space and time. The nonLipschitz condition is described by the behavior of the time-derivative of coefficients. This leads to a classification of oscillations, where fast oscillations represent the critical case. To study this critical case we propose a refined perfect diagonalization procedure basing on suitable zones of the phase space and corresponding nonstandard symbol classes. After this diagonalization procedure we construct the parametrix in several steps. Here the perfect diagonalization helps to understand, that only a finite loss of derivatives appears for solutions valued in Sobolev spaces. We point out where this loss comes from. From construction of parametrix we conclude a result about C ∞-well posedness of the Cauchy problem. †Dedicated to the memory of Professor Sigeru Mizohata.