Abstract:
We consider the highfrequency Helmholtz equation in
the whole space, with a variable index of
refraction. More precisely, we select the outgoing
solution to the abovementionned Helmholtz equation,
i.e. the (unique) solution that satisfies the
Sommerfeld radiation condition at infinity. Our main
result establishes that this solution radiates
energy in the outgoing direction only, uniformly
along the highfrequency parameter. This result
holds provided the associated rays of geometric
optics nicely escape to infinity, and they satisfy
an original nonfocusing condition. The analysis
relies on carefully following the dispersive
properties of the Helmholtz equation, for small,
moderate, and large values of the space variable. A
counterexample shows the optimality of the
nonfocusing condition. Uniform boundedness of the
solution is also considered.
[LECTURE SLIDES]
