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

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.

We prove uniform Morrey-Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted L 2 -estimates for the solution. We also prove a uniform L 2 -estimate without weight for the trace of the solution on the interface.

In this text, we compute the high frequency limit of the Hemholtz 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.

We study a model of magnetization switching driven by a spin current: the magnetization reversal can be induced without applying an external magnetic field. We first write our one-dimensional model in an adimensionalized form using a small parameter ε. We then explain the various time and space scales involved in the studied phenomena. Taking into account these scales, we first construct an appropriate numerical scheme that allows us to recover numerically various results of physical experiments. We then perform a formal asymptotic study as ε tends to 0 using a multiscale approach and asymptotic expansions. We thus obtain approximate limit models that we compare with the original model via numerical simulations.