Approximate solutions and micro-regularity in the Denjoy–Carleman classes

Abstract

We begin with a sequence M of positive real numbers and we consider the Denjoy-Carleman class C M . We show how to construct M-approximate solutions for complex vector fields with C M coefficients. We then use our construction to study micro-local properties of boundary values of approximate solutions in general M-involutive structures of codimension one, where the approximate solution is defined in a wedge whose edge (where the boundary value exists) is a maximally real submanifold. We also obtain a C M version of the Edge-of-the-Wedge Theorem.Published by Elsevier Inc.

Microlocal Regularity for Mizohata Type Differential Operators

“…It was then noticed that Hanges and Treves technique was very useful in connection with the existence of approximate solutions. The existence of approximate solutions in Denjoy-Carleman classes of functions in connection with problem arising in PDEs has been exhaustively studied in recent years (see, for instance [2][3][4][5]8]), where the authors used these approximate solutions to extend Asano's result to these classes of ultradifferentiable functions.…”

“…Using (2.1), we can further express L given by (3.1) as Proof. We will follow the ideas in [18,32,33] in connection with the proof of existence of approximate solutions in Denjoy-Carleman classes as given in [3]. To this end, let v be the approximate solution to the problem…”

“…The existence of such solutions v is guaranteed by [3] and it can be constructed by looking at formal power series of λ(x, t) and the formal solutions v(x, t), both in t = 0. Indeed, write…”

“…Now, using the sequence of cutoff functions (a k ) k∈N 0 introduced by Lambert, [24] (see also [3]) we can proceed as in [3] to write…”

On microlocal analyticity and smoothness of solutions of first-order nonlinear PDEs

“…Microlocal smoothness results for nonlinear PDEs were obtained in [3,9]. For results on Gevrey/Denjoy-Carleman regularity we refer the reader to [1,2,5]. The approach to the fully nonlinear case by using the Holomorphic Hamiltonian is motivated by [4].…”