High Frequency Analysis of Helmholtz Equations: Case of Two Point Sources

Abstract

Abstract. We derive the high frequency limit of the Helmholtz equation with source term when the source is the sum of two point sources. We study it in terms of Wigner measures (quadratic observables). We prove that the Wigner measure associated with the solution satisfies a Liouville equation with, as source term, the sum of the source terms that would be created by each of the two point sources taken separately. The first step, and main difficulty, in our study is the obtention of uniform estimates on the solution. Then, from these bounds, we derive the source term in the Liouville equation together with the radiation condition at infinity satisfied by the Wigner measure.

Mesures limites pour l’équation de Helmholtz dans le cas non captif

“…On a alors le théorème suivant quiétend le résultat de Fouassier [9] aux cas a coefficients variables. …”

“…Nous aurons besoin d'une hypothèse géométrique qui, comme précédemment, est essentiellement nécessaire pour l'unicité de la mesure. Fouassier [9] avaitétudié cette question sous l'hypothèse V = 0.…”

Semiclassical measure for the solution of the dissipative Helmholtz equation

“…Wang and P. Zhang give a proof for V 1 = 0 (variable refraction index) in [30] using uniform estimates given by Mourre method. We also mention the work of E. Fouassier who considers the case of a source which concentrates on two points (see [12], V 1 = 0 in this case) and the case of a potential discontinuous along an affine hyperplane in [13] (the source concentrates on 0 in this case). All these papers use a priori estimates of the solution (we have already mentioned [21], see also [6,30,29,7]).…”

Semiclassical measure for the solution of the Helmholtz equation with an unbounded source

“…This restriction was overcome in [20]. In [8] the source term concentrates on two points and in [9] the refraction index is discontinuous along an hyperplane of R n . All these papers study the semiclassical measure of the solution using its Wigner transform.…”