Introduction to the mathematical analysis of the Helmholtz equation : Sommerfeld condition, limiting amplitude principle and limiting absorption principle

Hélène Barucq 1, 2 Julien Diaz 1, 2 Sébastien Tordeux 1, 2
1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : The Helmholtz equation models time-harmonic wave motion phenomena and is consequently one of the most important equation in mathematical physics. For infinite domains, its mathematical analysis is rather difficult due to a default of coercivity of the associated operator: the solutions of the Helmholtz equation are not of finite energy and are not uniquely defined. Many authors have developed a theory to bypass these difficulties. The limiting amplitude technique, the absorbing principle and the unique continuation theorem are, to my opinion, the main ingredients of this theory. In this lecture, I will give an introduction to these three techniques.
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Hélène Barucq, Julien Diaz, Sébastien Tordeux. Introduction to the mathematical analysis of the Helmholtz equation : Sommerfeld condition, limiting amplitude principle and limiting absorption principle. Summer School Jaca 2012, Sep 2012, Jaca, Spain. ⟨hal-00927403⟩

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