Semiclassical measure for the solution of the dissipative Helmholtz equation

Julien Royer (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.

Mots clés: Dissipative operators; Helmholtz equation; Semiclassical measures

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Julien Royer. Semiclassical measure for the solution of the dissipative Helmholtz equation. Journal of Differential Equations, 2010, 249 (11), p. 2703-2756. ⟨10.1016/j.jde.2010.07.004⟩. ⟨hal-00434327⟩