Abstract
A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset $\Omega$ of $IR^n$, with initial conditions compactly supported in $\Omega$, and Dirichlet or Neumann type boundary condition. The transport of the microlocal energy density along the broken bicharacteristic flow at the high frequency limit is proved through the use of Wigner measures. Our approach consists first in computing explicitly the Wigner measures under an additional control of the initial data allowing to approach the solution by a superposition of first order Gaussian beams. The results are then generalized to standard initial conditions.
Highlights
We are interested in the high frequency limit of the initial-boundary value problem (IBVP) for the wave equation n
We are interested in the description of the behavior of the local energy density n c2|∂xj uε|2, at the high frequency limit ε → 0, in which case, it is j=1 well known that this quantity can be computed through the use of Wigner measures
We present an approach to compute Wigner measures based on the Gaussian beam formalism
Summary
We are interested in the high frequency limit of the initial-boundary value problem (IBVP) for the wave equation n. As the microlocal energy density of one individual beam is concentrated near its associated bicharacteristic, one would expect that the Wigner measure of a summation of weighted Gaussian beams will yield that the associated weights are transported along the broken bicharacteristic flow (see p.8 for the construction of reflected flows and p.29 for the definition of the broken flow) This result is not immediate as even different beams become infinitely close to each other. For ε-oscillatory initial data with Wigner measures not charging the set Rn × {ξ = 0}, a truncation of their frequency support at infinity and at zero does not affect the energy density of the solution as ε → 0.
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