Alberto Ruiz

The Mizohata-Takeuchi conjecture for the resolvent of the Navier-Lamé equation is a weighted estimate with weights in the so-called Mizohata-Takeuchi class for this operator when one approaches the spectrum (Limiting Absorption Principles). We prove this conjecture in dimensions 2 and 3 for weights with a radial majorant in the Mizohata-Takeuchi class. This result can be seen as an extension of the analogue for the Laplacian given in [8]. We also prove that radial weights in this class are not invariant for the Hardy-Littlewood maximal function, hence the methods in [6] used to extend estimates for the Laplacian to the Navier-Lamé case, do not work.

We present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator - Δ + q {-\Delta+q} , where q is a small real-valued potential with compact support in the Sobolev space W β , 2 {W^{\beta,2}} , with β > 0 . {\beta>0.} This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992, 55–68], in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension 4, but for more regular potentials in W β , 2 {W^{\beta,2}} , with β > 2 / 3 {\beta>2/3} . The proof is a consequence of the reconstruction method presented in our previous work, [J. A. Barceló, C. Castro, T. Luque and M. C. Vilela, A new convergent algorithm to approximate potentials from fixed angle scattering data, SIAM J. Appl. Math. 78 2018, 2714–2736].

We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of the Weyl tensor (in dimensions 4 and higher) and the Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set.