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Lectures on geometric optics. (English) Zbl 0926.35003

Caffarelli, Luis (ed.) et al., Hyperbolic equations and frequency interactions. Providence, RI: American Mathematical Society. IAS/Park City Math. Ser. 5, 383-466 (1999).
The main aim of these lecture notes is to make the reader familiar with some basic concepts and results related to hyperbolic systems of nonlinear partial differential equations which involve a small parameter \(\varepsilon\) (which denotes the wave-length normalized in an appropriate manner). The term geometric optics appearing in the title is used synonymously with the asymptotic analysis of solutions when \(\varepsilon\to 0\). The attention is concentrated on the so-called semilinear equations, namely: \[ L(y,\partial_y)u+ F(y,u)= f(y), \] where \(L\) is a symmetric hyperbolic operator while the nonlinear function \(F(y,u): \mathbb{R}^{1+d}\times \mathbb{C}^N\to \mathbb{C}^N\) has partial derivatives of all orders which are uniformly bounded on sets of the form \(\mathbb{R}^{1+d}\times K\), with \(K\subset \mathbb{C}^N\). Here \(y\in \mathbb{R}^{1+d}\) stands for the vector whose components are the time and space coordinates. The lectures are aimed at the level of graduate students who are familiar with partial differential equations but not with the basic concepts of geometric optics. So, a big part of the lectures is devoted to developing background which prepares the way for nonlinear geometric optics.
The author divides the lectures into 7 parts as follows: Basic linear existence theorems; Examples of propagation of singularities and energy; Elliptic geometric optics; Linear hyperbolic geometric optics; Basic nonlinear existence theorems; One phase nonlinear geometric optics; Justification of one phase nonlinear geometric optics.
The notes, which involve 35 lemmas, theorems and propositions, have a rather particular style: Main ideas of an argument or a proof or a special case are often described with their essential points while details and generalizations are given as exercises (more than 57) to be worked out by the reader. Research students in mathematics and theoretical physics as well as those in electrical engineering can find very important basic material of high level in these notes.
For the entire collection see [Zbl 0906.00018].

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
78A05 Geometric optics
35L60 First-order nonlinear hyperbolic equations
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