Séminaire d'analyse (archives)

We consider compact smooth Riemannian manifolds with boundary of dimension greater than two. We show that for wave equations, boundary data on the manifold is enough to determine time dependent and time independent lower order source terms in a variety of geometric settings. The main technique is the use of the Gaussian beam Ansatz. We briefly discuss the relationship of the work to recent progress on the Calderon problem.

We consider Sobolev spaces on Riemannian manifolds. A natural question to ask is whether these spaces are an algebra under the pointwise product as in the Euclidean case. We shall give sufficient conditions on the underlying Laplace-Beltrami operator to ensure such algebra properties, and describe implications for chain rules and paralinearisation formulas. Our results rely on abstract paraproducts defined via functional calculus for self-adjoint operators. This is joint work with F. Bernicot and T. Coulhon.