Opérateurs pseudo-différentiels et théorème de Nash-Moser
This book is a course on partial differential equations and pseudodifferential operators. It consists of three parts. The choice of material together with its arrangement shows some non-obvious links between seemingly distant topics of interest. The first part develops the theory of pseudodifferential operators. This generalization of partial differential operators is based on the fact that differentiation acts as polynomial multiplication in the Fourier regime. If we use a (reasonable) non-polynomial function as the multiplier, we obtain a pseudodifferential operator. The multiplier may depend on the initial space variable. Following this the case of variable coefficients is covered. The objective here is to establish the most important formulas of the ensuing calculus.
The second part presents the Littlewood-Paley theory and microlocal analysis (in particular a concept of the wave front set of a distribution in connection with pseudodifferential operators, energy estimates and propagation of singularities). In comparison with the first chapter, the exposition moves towards more specific problems motivated by partial differential equations but still in the pseudodifferential setting.
The third part studies perturbations of problems in partial differential equations. Starting with applications of the implicit function theorem and going through the situation treated by fixed point theorems, the main goal of the chapter is the Nash-Moser theorem. The results describe existence problems and estimates for solutions of perturbed problems. The main difficulty is to handle the loss of derivatives appearing when solving the linearized problem. The Nash theorem on existence of an isometric embedding of a Riemannian manifold is included as a special case. The book is intended as a course for advanced students but it will also be very useful for researchers. The material contained here is deep and very important for the understanding of some issues of the theory of partial differential equations and the more general context of pseudodifferential operators. The presentation is quite compact and the student should be well prepared (in particular, a good knowledge of Fourier calculus is needed). When the authors say “elementary” it should sometimes be read as “short”. The exposition is highly self-contained and the text is complemented by numerous exercises.