Oliver Lindblad Petersen: Unique continuation through compact hypersurfaces

Abstract.

Assume that a solution to a linear homogeneous partial differential equation vanishes identically on one side of a hypersurface in Euclidean space or in a more general manifold. Does it necessarily vanish also on the other side of the hypersurface? This is known as the question of unique continuation. In this talk, I will explain two classical unique continuation theorems by Calderón and Hörmander in the case of linear wave equations. However, these are insufficient for important rigidity questions in cosmology. Fortunately, assuming that the hypersurface is compact, unique continuation is true under much weaker assumptions than those of Calderón or Hörmander. I will describe this phenomenon by presenting simple explicit examples and counterexamples and explain the proof in a special case.