High Frequency limit of the Helmholtz Equations

Abstract

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of L 2 bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.

Mathematical tools for kinetic equations

“…This kind of problem arises naturally in kinetic physics (for instance flows around an obstacle share the same mathematical aspects) and in the high frequency limit of dispersive equations; see [100], [59], [17] and the references therein. We consider equation (14) ξ…”

Morrey–Campanato estimates for Helmholtz equations with two unbounded media

“…The condition (H5) involves both the interface and the index: it becomes a weaker assumption on the index when the interface is close to a hyperplane (α ∼ 1). This type of assumptions is natural in the study of the high frequency limit where the link with Liouville's equations can be understood through Wigner transform (see L. Miller [8] for a refraction result in the case of a sharp interface for Schrodinger equation, E. Fouassier [6] for high frequency limit of Helmholtz equations with interface; for an account on high frequency limit for wave equations, see P.-L. Lions, T.Paul [7], and [2], [3] for Helmholtz equations without interface).…”

“…First, the homogeneity of the estimates and assumptions makes this theorem compatible with the high frequencies. The scaling invariance plays a fundamental role in the high frequency limit of Helmholtz equations (see Benamou et al [2], Castella et al [3] for the case of a regular index of refraction).…”

Mathematical models for laser-plasma interaction

“…[3][4][5][6] and [38] (Chaps. 1 and 5) for a physical presentation of the above derivation, particularly for a justification of formula (14) of the plasma conductivity.…”

“…The derivation of this model has been well known for a long time; see [25,29] for example and [31] for a more recent presentation. For a rigorous presentation of the geometrical optics approximation in the case of a source located on a manifold, see [5,13]. For a numerical treatment by a classical ray tracing method, see for example [43]; and by Eulerian methods see [4,6,7,44].…”