European Mathematical Society - edp sciences
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enOpérateurs pseudo-différentiels et théorème de Nash-Moser
https://euro-math-soc.eu/review/op%C3%A9rateurs-pseudo-diff%C3%A9rentiels-et-th%C3%A9or%C3%A8me-de-nash-moser
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The book is based on an advanced university lecture course providing an extensive study of basic properties of pseudo-differential operators and some of their principal applications, with particular emphasis on the Nash-Moser theorem. The exposition is written in a very attractive way, which should appeal to experts in the field as well as to PhD students (or even gifted undergraduates) with some basic knowledge of elementary functional analysis, Fourier analysis and, perhaps, the theory of distributions. The book is self-contained, and the authors have taken a lot of trouble to make their exposition as reader-friendly as possible.<br />
The book is divided into three chapters (plus an introductory Chapter 0). Chapter I is an exposition of the theory of pseudo-differential operators (the authors call this part a ‘minimal theory’, but it is in fact quite comprehensive). The material includes the concept of a symbol, its use in operator calculus, the action of operators on Sobolev spaces and the invariance under change of variables. Chapter II is divided into three themes, covering (among other topics) the Littlewood-Paley theory of dyadic decomposition of distributions (with such interesting facts as a characterisation of the Hölder and Sobolev spaces), ‘micro-local analysis’ and some energy estimates. The last chapter, ‘The implicit function theorems’ treats the role of implicit functions in elliptic problems, examples of applications of fixed-point theorems to semilinear hyperbolic systems, and a thorough and comprehensive exposition of the Nash-Moser theorem on the existence and properties of a solution to the equation Φ(u) = Φ(u0) + f.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">lp</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-alinhac" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. alinhac</a></li><li class="vocabulary-links field-item odd"><a href="/author/p-g%C3%A9rard" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p. gérard</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/edp-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">edp sciences</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2000</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-222-04535-7</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">FRF 230</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/46-functional-analysis" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46 Functional analysis</a></li></ul></span>Wed, 15 Jun 2011 20:33:19 +0000Anonymous39518 at https://euro-math-soc.euOpérateurs pseudo-différentiels et théorème de Nash-Moser
https://euro-math-soc.eu/review/op%C3%A9rateurs-pseudo-diff%C3%A9rentiels-et-th%C3%A9or%C3%A8me-de-nash-moser-0
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>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. </p>
<p>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. </p>
<p>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.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jama</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/s-alinhac" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">s. alinhac</a></li><li class="vocabulary-links field-item odd"><a href="/author/p-g%C3%A9rard" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p. gérard</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/edp-sciences" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">edp sciences</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2000</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">2-222-04535-7 or 2-7296-0364-6</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">EUR 35</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Wed, 08 Jun 2011 11:27:13 +0000Anonymous39386 at https://euro-math-soc.eu