Section: New Results

Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

In [43] , we consider the high frequency Helmholtz equation with a variable refraction index $n^2\left(x\right)$ ($x\in {ℝ}^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter ${\alpha }_{\epsilon }>0$ is added, which prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\epsilon >0$. We let $\epsilon$ and ${\alpha }_{\epsilon }$ go to zero simultaneously. We study the question whether the prescribed radiation condition at infinity is satisfied uniformly along the asymptotic process $\epsilon \to 0$. This question has been previously studied by the first author, who has proved that the radiation condition is indeed satisfied uniformly in $\epsilon$, provided the refraction index satisfies a specific non-refocusing condition. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again. In the present text we show the optimality of the above mentioned non-refocusing condition. We exhibit a refraction index which does refocus the rays of geometric optics sent from the origin near the origin again, and we show that the limiting solution does not satisfy the natural radiation condition at infinity in that case.