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Carlos Kenig

Publications and source records attributed to Carlos Kenig.

At least 19 recordsLinked to original sources
arXiv2024

A note on the critical set of harmonic functions near the boundary

Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,\alpha}$ domain for some $\alpha \in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 = |\nabla u(x)| \}$ (see [KZ1, KZ2]).

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arXiv2022

Examples of non-Dini domains with large singular sets

Let $u$ be a non-trivial harmonic function in a domain $D\subset \mathbb{R}^d$ which vanishes on an open set of the boundary. In a recent paper, we showed that if $D$ is a $C^1$-Dini domain, then within the open set the singular set of $u$, defined as $\{X\in \overline{D}: u(X) = 0 = |\nabla u(X)|\} $, has finite $(d-2)$-dimensional Hausdorff measure. In this paper, we show that the assumption of $C^1$-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose \textit{singular sets} have infinite $\mathcal{H}^{d-2}$-measures.

math.CV↗
arXiv2022

Soliton resolution for the radial quadratic wave equation in six space dimensions

We consider the quadratic semilinear wave equation in six dimensions. This energy critical problem admits a ground state solution, which is the unique (up to scaling) positive stationary solution. We prove that any spherically symmetric solution, that remains bounded in the energy norm, evolves asymptotically to a sum of decoupled modulated ground states, plus a radiation term. As a by-product of the approach we prove the non-existence of multisoliton solutions that do not emit any radiation. The proof follows the method initiated for large odd dimensions by the last three authors, reducing the problem to ruling out the existence of such non-radiative multisolitons, by deriving a contradiction from a finite dimensional system of ordinary differential equations governing their modulation parameters. In comparison, the difficulty in six dimensions is the failure of certain channel of energy estimates and the related existence of a linear resonance. The main novelties are the use of new channel of energy estimates, as well as the classification of non-radiative solutions with small energy.

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arXiv2022

On channels of energy for the radial linearised energy critical wave equation in the degenerate case

Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation they have been obtained in the radial case in odd dimensions, first in $3$ dimensions by Duyckaerts, Kenig and Merle (Camb. J. Math., 2013), then for general odd dimensions by the same authors (Comm. Math. Phys., 2020). We consider even dimensions, for which such estimates are known to fail (Côte, Kenig and Schlag, Math. Ann., 2014). We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us to prove the soliton resolution conjecture in six dimensions (Collot, Duyckaerts, Kenig and Merle, arXiv preprint 2201.01848, 2022 versions 1 and 2).

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arXiv2022

On classification of non-radiative solutions for various energy-critical wave equations

Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region $|x|>R+|t|$ vanishes asymptotically in both time directions. This notion, introduced by Duyckaerts, Kenig and Merle (J. Eur. Math. Soc., 2011), has been key in solving the soliton resolution conjecture for these equations in the radial case. In the present paper, we first classify their asymptotic behaviour at infinity, showing that they correspond to a $k$-parameters family of solutions where $k$ depends on the dimension. This generalises the previous results (Duyckaerts, Kenig and Merle, Camb. J. Math., 2013 and Duyckaerts, Kenig, Martel and Merle, Comm. Math. Phys., 2022) in three and four dimensions. We then establish a unique maximal extension of these solutions.

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arXiv2021

Expansion of harmonic functions near the boundary of Dini domains

Let $u$ be a harmonic function in a $C^1$-Dini domain, such that $u$ vanishes on an open set of the boundary. We show that near every point in the open set, $u$ can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that $u$ has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.

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arXiv2021

Soliton resolution for critical co-rotational wave maps and radial cubic wave equation

In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.

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arXiv2021

Boundary unique continuation on $C^1$-Dini domains and the size of the singular set

Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

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arXiv2020

Improved quantitative unique continuation for complex-valued drift equations in the plane

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form $Δu + W \cdot \nabla u = 0$ in $\mathbb{R}^2$, where $W = W_1 + i W_2$ with each $W_j$ real-valued. Under the assumptions that $W_j \in L^{q_j}$ for some $q_1 \in [2, \infty]$, $q_2 \in (2, \infty]$, and $W_2$ exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.

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arXiv2019

Propagation of smallness in elliptic periodic homogenization

The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_ε$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.

math.AP↗
arXiv2019

The focusing energy-critical nonlinear wave equation with random initial data

We consider the focusing energy-critical quintic nonlinear wave equation in three dimensional Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_x(\mathbb{R}^3)$, for any $s > 1/2$. By randomizing radial initial data in $ \dot H^s_x(\mathbb{R}^3) \times H^{s-1}_x(\mathbb{R}^3)$ for $s > 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton which give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the first long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

math.AP↗
arXiv2019

Characterization of large energy solutions of the equivariant wave map problem: I

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a rescaling. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q). The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blow-up constructions for degree one, 1-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Raphael and Rodnianski.

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arXiv2018

On Landis' conjecture in the plane when the potential has an exponentially decaying negative part

In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis' conjecture by establishing unique continuation at infinity estimates for solutions to equations of the form $- Δu + V u = 0$ in $\mathbb{R}^2$, where $V = V_+ - V_-$, $V_+ \in L^\infty$, and $V_-$ is a non-trivial function that exhibits exponential decay at infinity. The main tool in the proof of this theorem is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing estimate, we establish a similarity principle for vector-valued Beltrami systems.

math.AP↗
arXiv2017

Construction of 2-peakon Solutions and Ill-Posedness for the Novikov equation

For the Novikov equation, on both the line and the circle, we construct a 2-peakon solution with an asymmetric antipeakon-peakon initial profile whose $H^s$-norm for $s<3/2$ is arbitrarily small. Immediately after the initial time, both the antipeakon and peakon move in the positive direction, and a collision occurs in arbitrarily small time. Moreover, at the collision time the $H^s$-norm of the solution becomes arbitrarily large when $5/4 3/2$, these results put together establish $3/2$ as the critical index of well-posedness for this equation. The case $s=3/2$ remains an open question.

math.AP↗
arXiv2017

Soliton resolution along a sequence of times for the focusing energy critical wave equation

In this paper, we prove the soliton resolution conjecture for general type II solutions to the focusing energy critical wave equation, in space dimension 3,4 or 5, along a sequence of times. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. This paper is an extension of the arXiv preprint 1510:00075 by the second author, where the finite time blow-up case is treated, with an error converging to zero in a weaker sense.

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arXiv2016

Universality of blow up profile for small blow up solutions to the energy critical wave map equation

In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that for a wave map that has energy just above the degree one harmonic maps and that blows up in finite time, the solution asymptotically de-couples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called quantization of energy at blow up. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps.

math.AP↗
arXiv2016

$L^p$ resolvent estimates for magnetic Schrödinger operators with unbounded background fields

We prove $L^p$ and smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials.

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