Geometric Optics Expansions with Amplification for Hyperbolic Boundary Value Problems: Linear Problems

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that lossesof derivatives must occur from the source terms to the solution. Secondly, we are able to derive a lower bound forthe finite speed of propagation, showing that waves may propagate faster than for the propagation in free space. We illustrate our analysis with some examples.