David Dos Santos Ferreira

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderón problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.

Determining an unbounded potential from Cauchy data in admissible geometries

In [4] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L n/2. In the process, we derive L p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8] and Kenig et al. [9].

On Lp resolvent estimates for Laplace Beltrami operators on compact manifolds

Abstract. In this article we prove Lp estimates for resolvents of Laplace–Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge (1987) in the Euclidean case and Shen (2001) for the torus. We follow Sogge (1988) and construct Hadamards parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sjölin condition. Our initial motivation was to obtain Lp Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig (1985); we illustrate the pertinence of Lp resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the construction of complex geometrical optics solutions to the Schrödinger equation with unbounded potentials, an essential device for solving anisotropic inverse problems.

Global and local regularity of Fourier integral operators on weighted and unweighted spaces

We investigate the global continuity on

Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map

L^p

The Linearized Calderón Problem in Transversally Anisotropic Geometries

spaces with

Strichartz Estimates for Non-selfadjoint Operators and Applications

p\in [1,\infty]

Determining a magnetic Schrödinger operator from partial cauchy data

of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain non-degeneracy conditions. We initiate the investigation of the continuity of smooth and rough Fourier integral operators on weighted

The Calderón problem in transversally anisotropic geometries

L^{p}

Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map

spaces,