Sommerfeld condition for a Liouville equation and concentration of trajectories

Abstract

We analyse the concentration of trajectories in a Liouville equation set in the full space with a potential which is not constant at infinity. Our motivation comes from geometrical optics where it appears as the high freqency limit of Helmholtz equation. We conjecture that the mass and energy concentrate on local maxima of the refraction index and prove a result in this direction. To do so, we establish a priori estimates in appropriate weighted spaces and various forms of a Sommerfeld radiation condition for solutions of such a stationary Liouville equation.