Penelope Gehring: Finite propagation speed vs. non-local boundary conditions

The question of finite propagation speed is fundamental in the study of linear wave equations. It asserts that a wave can propagate with at most the speed of light. In this talk, we focus on Cauchy problems for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds, both with and without timelike boundary. We will recall some finite propagation speed results for the boundaryless case as well as for local boundary conditions. 

We then turn to the case of non-local boundary conditions, where an interesting phenomenon occurs: when a wave first encounters the boundary, the entire boundary instantaneously radiates back everywhere. This allows signals to propagate faster than with any prescribed velocity. We will see that this behavior emerges directly from how non-locality influences the energy estimates. Although it may initially appear to be a limitation of the method, we will confirm with explicit examples that this effect really occurs.