Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

摘要

Given a Gs-involutive structure, (M,V), a Gevrey submanifold X?M which is maximally real and a Gevrey function u0 on X we construct a Gevrey function u which extends u0 and is a Gevrey approximate solution for V. We then use our construction to study Gevrey micro-local regularity of solutions, u∈C2(RN), of a system of nonlinear pdes of the form. Fj(x,u,ux)=0,j=1,. .,n, where Fj(x,ζ0,ζ) are Gevrey functions of order s>1 and holomorphic in (ζ0,ζ)∈C×CN. The functions Fj satisfy an involutive condition and dζF1^...^dζFn≠0.