Members & Former Members

We study the Cauchy problem for symmetric hyperbolic systems on Lorentzian manifolds with timelike boundary, where initial values are prescribed on a spacelike slice and nonlocal boundary conditions are imposed on the timelike boundary. This extends the classical theory of local boundary conditions to include matching conditions and slicewise Atiyah--Patodi--Singer type boundary conditions.We prove existence of weak solutions and, under suitable regularity assumptions on the Cauchy data, establish existence and uniqueness of strong solutions.

An interesting phenomenon occurs for finite propagation speed: for nonlocal boundary conditions, 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.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds