High frequency limit of Helmholtz equations: refraction by sharp interfaces

Abstract

International audienceIn this paper, we compute the high frequency limit of the Helmholtz equation with source term, in the case of a refraction index that is discontinuous along a sharp interface between two unbounded media. The asymptotic propagation of energy is studied using Wigner measures. Our result is twofold. First, in the general case, assuming some geometrical hypotheses on the index and assuming that the interface does not capture energy asymptotically, we prove that the limiting Wigner measure satisfies a stationary transport equation with source term. As a consequence, the Wigner measure is characterized as the integral, along the rays of geometrical optics and up to infinite time, of the energy source. This result encodes the refraction phenomenon. Second, we study the particular case when the index is constant in each media, for which the analysis goes further: we prove that the interface does not capture energy asymptotically in this case

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

“…Lorsque la source est portée par une variété, des résultatséquivalents ontété obtenus par Castella, Perthame et Runborg [6] dans le cas V = 0 et généralisés par Wang et Zhang [26] aux casà coefficients variables. Enfin, quand le potentiel est constant de chaque coté d'un hyperplan sur lequel il présente une discontinuité, la propagation de telles mesures semi-classiques aété traitée par Fouassier [10]. Tous ces travaux utilisent des majorations de la résolvante entre espaces de Besov (estimations de Morrey-Campanato).…”

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.…”

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]).…”