On Microlocal Smoothness of Solutions of First Order Nonlinear PDE

Abstract

. We study the microlocal smoothness of C 2 solutions u of the first-order nonlinear partial differential equation x where f ( x,t,ζ 0 ,ζ ) is a complex-valued function which is C ∞ in all the variables ( x,t,ζ 0 ,ζ ) and holomorphic in the variables ( ζ 0 ,ζ ) . If the solution u is C 2 , σ ∈ Char( L u ) and 1 √− 1 σ ([ L u , ¯ L u ]) < 0 , then we show that σ / ∈ WF ( u ) . Here WF ( u ) denotes the C ∞ wave front set of u and Char( L u ) denotes the characteristic set of the linearized operator