38 Result(s)

Chapter

On the Work of Jean Bourgain in Nonlinear Dispersive Equations

In this brief note, we survey a sample of the deep and influential contributions of Jean Bourgain to the field of nonlinear dispersive equations. Bourgain also made many fundamental contributions to other area...

Carlos E. Kenig in Analysis at Large (2022)

Article

Energy partition for the linear radial wave equation

We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy, as time goes to infinity. We show that the exterior energy estimate, which Duyc...

Raphaël Côte, Carlos E. Kenig, Wilhelm Schlag in Mathematische Annalen (2014)

Article

Relaxation of Wave Maps Exterior to a Ball to Harmonic Maps for All Data

In this paper we establish relaxation of an arbitrary 1-equivariant wave map from $${\mathbb{R}^{1+3}_{t,x}{\setminus} (\mathbb{R}\tim...

Carlos E. Kenig, Andrew Lawrie, Wilhelm Schlag in Geometric and Functional Analysis (2014)

Article

Convergence Rates in L ² for Elliptic Homogenization Problems

We study rates of convergence of solutions in L ² and H ^1/2 for a family of elliptic systems ...

Carlos E. Kenig, Fanghua Lin, Zhongwei Shen in Archive for Rational Mechanics and Analysis (2012)

Chapter

The Concentration-Compactness Rigidity Method for Critical Dispersive and Wave Equations

In these lectures I will describe a program (which I will call the concentrationcompactness/rigidity method) that Frank Merle and I have been develo** to study critical evolution problems. The issues studied...

Carlos E. Kenig in Nonlinear Partial Differential Equations (2012)

Article

Homogenization of elliptic boundary value problems in Lipschitz domains

In this paper we study the L ^p boundary value problems for $${\mathcal{L}(...

Carlos E. Kenig, Zhongwei Shen in Mathematische Annalen (2011)

Article

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believ...

Shangbin Cui, Carlos E. Kenig in Journal of Fourier Analysis and Applications (2010)

Chapter

Recent Progress on the Global Well-Posedness of the KPI Equation

In this chapter I survey the well-posedness theory for the Kadomtsev–Petviashvili (KPI) equation, culminating with the recent proof (joint with A. Ionescu and D. Tataru) of the global well-posedness of the KPI...

Carlos E. Kenig in Recent Developments in Real and Harmonic Analysis (2010)

Article

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 Eucl...

David Dos Santos Ferreira, Carlos E. Kenig, Mikko Salo… in Inventiones mathematicae (2009)

Article

Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which...

Carlos E. Kenig, Frank Merle in Acta Mathematica (2008)

Article

Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

Given g and f  =  gg′, we consider solutions to the following non linear wave equation : $$\left\{ \begin{array}{l}\displaystyle u_{tt...

Raphaël Côte, Carlos E. Kenig, Frank Merle in Communications in Mathematical Physics (2008)

Article

Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrödinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determin...

David Dos Santos Ferreira, Carlos E. Kenig… in Communications in Mathematical Physics (2007)

Article

Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥  3

In dimensions d ≥ 3, we prove that the Schrödinger map initial-value problem $$ \left\{ \begin{array}{l} \partial_ts=s\times\Delta_x s...

Alexandru D. Ionescu, Carlos E. Kenig in Communications in Mathematical Physics (2007)

Article

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

Carlos E. Kenig, Frank Merle in Inventiones mathematicae (2006)

Article

On localization in the continuous Anderson-Bernoulli model in higher dimension

Jean Bourgain, Carlos E. Kenig in Inventiones mathematicae (2005)

Article

The Cauchy problem for quasi-linear Schrödinger equations

Carlos E. Kenig, Gustavo Ponce, Luis Vega in Inventiones mathematicae (2004)

Article

L ^p Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations

Alexandru D. Ionescu, Carlos E. Kenig in Acta Mathematica (2004)