The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave packet approach

Castella, François

Journées équations aux dérivées partielles (2004), article no. 4, 18 p.

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\alpha >0$. The high-frequency (or: semi-classical) parameter is $\epsilon >0$. We let $\epsilon$ and $\alpha$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.

Under these assumptions, we prove that the solution $u^{\epsilon }$ radiates in the outgoing direction, uniformly in $\epsilon$. In particular, the function $u^{\epsilon }$, when conveniently rescaled at the scale $\epsilon$ close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform (in $\epsilon$) version of the limiting absorption principle.

Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) For very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in $\epsilon$.

Classification : 35Q40, 35J10, 81Q20

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