Strong Gevrey Solvability for a System of Linear Partial Differential Equations

Abstract

We consider a class of linear systems whose principal symbol satisfies a certain condition of semi-hyperbolicity, and we prove the local sUljectivity in suitable Gevrey spaces. o IntroductionWe investigate the local solvability for m x m systems of typeHere u(t, x) and f(t, x) are em-valued functions on R l+n, while the A j (t, x)'s are m x m matrix functions, uniformly analytic in R l+n in the sense that there is some Co > 0 for which la:a~Aj(t, x)1~C6+We denote by t"1 (t , x,~), ... , t"m (t, x,~) the eigenvalues of the matrixrepeated following their multiplicities, and defineWe do not assume t"h 1= t"k for h 1= k, hence we cannot apply the classical theory of Coo-solvability of Hormander, Nirenberg and Treves for equations of principal type. Consequently, we expect solvability only in suitable Gevrey classes.Our main assumption is that, for each fixed~E R n , the imaginary parts of all the t"h'S keep the same sign when (t, x) runs in R l+n, that is