High-frequency limit of the Helmholtz equation with variable refraction index

We study the high-frequency limit of the Helmholtz equation with variable refraction index and a source term concentrated near a p-dimensional affine subspace. Under some conditions, we first derive uniform estimates in Besov spaces for the solutions. Then, we prove that the semi-classical measure associated with these solutions satisfies the stationary Liouville equation with an explicit source term and has certain radiation property at infinity. © 2005 Elsevier Inc. All rights reserved.