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C. E. Kenig

Publications and source records attributed to C. E. Kenig.

At least 19 recordsLinked to original sources
arXiv2024

On the fractional Schödinger equation with variable coefficients

We study the initial value problem (IVP) associated to the semi-linear fractional Schödinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

math.AP↗
arXiv2015

Hardy Uncertainty Principle, Convexity and Parabolic Evolutions

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.

math.AP↗
arXiv2010

The Hardy Uncertainty Principle Revisited

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.

math.AP↗
arXiv2009

Uncertainty Principle of Morgan type and Schrödinger Evolutions

We prove unique continuation properties for solutions of evolution Schrödinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an application of our method we also obtain results concerning the possible concentration profiles of solutions of semi-linear Schrödinger equations.

math.AP↗
arXiv2009

The sharp Hardy Uncertainty Principle for Schödinger evolutions

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schrödinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.

math.AP↗
arXiv2009

On the linearized local Calderon problem

In this article, we investigate a density problem coming from the linearization of Calderón's problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain $Ω$ vanishing on any fixed closed proper subset of the boundary are dense in $L^{1}(Ω)$ in all dimensions $n \geq 2$. This is proved using ideas coming from the proof of Kashiwara's Watermelon theorem.

math.AP↗
arXiv2008

Asymptotic stability of solitons for the Benjamin-Ono equation

In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of Martel and Merle (arXiv:0706.1174v2). As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.

math.AP↗
arXiv2008

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 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 X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.

math.AP↗
arXiv2008

Hardy's Uncertainty Principle, Convexity and Schrödinger Evolutions

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

math.AP↗
arXiv2007

Weighted Low-Regularity Solutions of the KP-I Initial Value Problem

In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper " Low regularity solutions for the Kadomstev-Petviashvili I equation " by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper.

math.AP↗
arXiv2005

The Calderón problem with partial data

In this paper we improve an earlier result by Bukhgeim and Uhlmann, by showing that in dimension larger than or equal to three, the knowledge of the Cauchy data for the Schrödinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of Bukhgeim and Uhlmann but use a richer set of solutions to the Dirichlet problem.

math.AP↗