# Carlos E. Kenig's research while affiliated with University of Chicago and other places

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## Publications (335)

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 measu...

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. I...

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...

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 three dimensions in [8], then for the general case in [10]. We consider even dimensions, for which such estimate...

Let u be a harmonic function in a C1-Dini domain D⊂Rd such that u vanishes on a boundary surface ball ∂D∩B5R(0). We consider an effective version of its singular set (up to boundary) S(u):={X∈D¯:u(X)=|∇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...

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 Δ u + W ⋅ ∇ u = 0 in ℝ 2 , where W = W 1 + i W 2 with each W j being real-valued. Under the assumptions that W j ∈ L q j for some q 1 ∈ [ 2 , ∞ ] , q 2 ∈ ( 2 , ∞ ] and t...

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 areas of partial differential equations and mathematical physics (as well as to a myriad of other areas in analysis, number theory, combinatorics...

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...

We consider the wave equation with an energy-supercritical focusing nonlinearity in dimension seven. We prove that any radial solution that remains bounded in the critical Sobolev space is global and scatters to a linear solution.

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...

This article is a memorial tribute to Richard L. Wheeden who made fundamental contributions to Harmonic Analysis, Potential Theory and Partial Differential equations. A short biographical sketch is also provided along with a description of the most important results obtained by Wheeden by himself and along with his collaborators, Richard Hunt and B...

We prove explicit doubling inequalities and obtain uniform upper bounds (under (d−1)-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combi...

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...

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...

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 onl...

We consider the energy-critical focusing wave equation in space dimension \(N\ge 3\). The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated Ws, de...

We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a com...

In this article, we survey recent results (with Duyckaerts and Merle) on the long-time behavior of radial solutions of the energy critical nonlinear wave equation in odd dimensions.

By definition, the exterior asymptotic energy of a solution to a wave equation on \({\mathbb {R}}^{1+N}\) is the sum of the limits as \(t\rightarrow \pm \infty \) of the energy in the the exterior \(\{|x|>|t|\}\) of the wave cone. In our previous work Duyckaerts et al. (J Eur Math Soc 14(5):1389–1454, 2012), we have proved that the exterior asympto...

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 $\Delta u + W \cdot \nabla u = 0$ 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, \i...

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if u1,u2 are two suitable solutions of the equation defined in Rn×[0,T] such that for some non-empty open set Ω⊂Rn×[0,T],u1(x,t)=u2(x,t) for (x,t)∈Ω, then u1(x,t)=u2(x,t) for any (x,t)∈Rn×[0,T]. The proof is b...

We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if $u_1,\,u_2$ are two suitable solutions of the equation defined in $\mathbb R^n\times[0,T]$ such that for some non-empty open set $\Omega\subset \mathbb R^n\times[0,T]$, $u_1(x,t)=u_2(x,t)$ for $(x,t) \in \O...

Consider the energy-critical focusing wave equation in odd space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. In this paper we prove that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, decoupled by th...

By definition, the exterior asymptotic energy of a solution to a wave equation on $\mathbb{R}^{1+N}$ is the sum of the limits as $t\to \pm\infty$ of the energy in the the exterior $\{|x|>|t|\}$ of the wave cone. In our previous work (JEMS 2012, arXiv:1003.0625), we have proved that the exterior asymptotic energy of a solution of the linear wave equ...

Consider the energy-critical focusing wave equation in space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation behaves asymptotically as a sum of modulated $W$s, dec...

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 $\mathcal{L}_\epsilon$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theo...

We consider the focusing energy-critical quintic nonlinear wave equation in 3D 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 in...

We prove that if u1,u2 are real solutions of the Benjamin-Ono equation defined in (x,t)∈R×[0,T] which agree in an open set Ω⊂R×[0,T], then u1≡u2. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-loc...

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 r...

We prove that if $u_1,\,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)\in\R \times [0,T]$ which agree in an open set $\Omega\subset \R \times [0,T]$, then $u_1\equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary valu...

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 $- \Del...

Consider a finite energy radial solution to the focusing energy critical
semilinear wave equation in 1+4 dimensions. Assume that this solution exhibits
type-II behavior, by which we mean that the critical Sobolev norm of the
evolution stays bounded on the maximal interval of existence. We prove that
along a sequence of times tending to the maximal...

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. Moreo...

In this paper, we prove that any solution of the energy-critical wave equation in space dimensions 3, 4 or 5, which is bounded in the energy space decouples asymptotically, for a sequence of times going to its maximal time of existence, as a finite sum of modulated solitons and a dispersive term. This is an important step towards the full soliton r...

We survey recent results related to soliton resolution.

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...

In this paper we study the global unique continuation property for the elasticity system and the general second-order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand,...

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...

This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((\beta,\infty))$ for...

Consider the focusing energy-critical wave equation in space dimension 3, 4
or 5. We prove that any global solution which is bounded in the energy space
converges in the exterior of wave cones to a radiation term which is a solution
of the linear wave equation.

In this work, we study the Landis conjecture for second-order elliptic
equations in the plane. Precisely, assume that $V\ge 0$ is a measurable
real-valued function satisfying $\|V\|_{L^\infty({\mathbb R}^2)} \le 1$. Let
$u$ be a real solution to $\mbox{div}(A \nabla u) - V u = 0$ in ${\mathbb
R}^2$. Assume that $|u(z)| \le \exp(c_0 |z|)$ and $u(0)...

In this paper, we give an overview of the authors' work on applications of
the method of concentration-compactness to global well-posedness, scattering,
blow-up and universal profiles for the energy critical wave equation in the
non-radial setting. New results and proofs are also given.

We prove $L^p$ and smoothing estimates for the resolvent of magnetic
Schr\"odinger 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
Schr\"odinger and Pauli operators with complex ele...

We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high
regularity for a class of nonlinear higher-order dispersive equations
generalizing the KdV hierarchy both on the line and on the torus.

Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument ba...

We prove that every solution of the focusing energy-critical wave equation
with the compactness property is global. We also give similar results for
supercritical wave and Schr\"odinger equations.

This paper is devoted to the proof of Lipschitz regularity, down to the
microscopic scale, for solutions of an elliptic system with highly oscillating
coefficients, over a highly oscillating Lipschitz boundary. The originality of
this result is that it does not assume more than Lipschitz regularity on the
boundary. Our Theorem, which is a significa...

In this paper we give a unified proof to the soliton resolution conjecture
along a sequence of times, for the semilinear focusing energy critical wave
equations in the radial case and two dimensional equivariant wave map
equations, including the four dimensional radial Yang Mills equation, without
using outer energy type inequalities. Such inequali...

In this paper, we provide a new means of establishing solvability of the
Dirichlet problem on Lipschitz domains, with measurable data, for second order
elliptic, non-symmetric divergence form operators. We show that a certain
optimal Carleson measure estimate for bounded solutions of such operators
implies a regularity result for the associated ell...

In this paper we consider finite energy, \ell-equivariant wave maps from
1+3-dimensional Minkowski space exterior to the unit ball at the origin into
the 3-sphere. We impose a Dirichlet boundary condition at r=1, which in this
context means that the boundary of the unit ball in the domain gets mapped to
the north pole. Each such \ell-equivariant wa...

Exterior channel of energy estimates for the radial wave equation were first
considered in three dimensions by Duyckaerts, the first author, and Merle, and
recently for the 5-dimensional case by the first, second, and fourth authors.
In this paper we find the general form of the channel of energy estimate in all
odd dimensions for the radial free w...

For a family of second-order elliptic operators with rapidly oscillating
periodic coefficients, we study the asymptotic behavior of the Green and
Neumann functions, using Dirichlet and Neumann correctors. As a result we
obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann
maps as well as near optimal convergence rates in $W^...

In this paper we derive quantitative uniqueness estimates at infinity for
solutions to an elliptic equation with unbounded drift in the plane. More
precisely, let $u$ be a real solution to $\Delta u+W\cdot\nabla u=0$ in
${\mathbf R}^2$, where $W$ is real vector and $\|W\|_{L^p({\mathbf R}^2)}\le K$
for $2\le p<\infty$. Assume that $\|u\|_{L^{\infty...

This paper is devoted to the proof of uniform H\"older and Lipschitz
estimates close to oscillating boundaries, for divergence form elliptic systems
with periodically oscillating coefficients. Our main point is that no structure
is assumed on the oscillations of the boundary. In particular, those are
neither periodic, nor quasiperiodic, nor station...

In this paper we prove a quantitative form of Landis' conjecture in the
plane. Precisely, let $W(z)$ be a measurable real vector-valued function and
$V(z)\ge 0$ be a real measurable scalar function, satisfying
$\|W\|_{L^{\infty}({\mathbf R}^2)}\le 1$ and $\|V\|_{L^{\infty}({\mathbf
R}^2)}\le 1$. Let $u$ be a real solution of $\Delta u-\nabla(Wu)-Vu...

Consider the focusing energy-critical wave equation in space dimension 3, 4
or 5. In a previous paper, we proved that any solution which is bounded in the
energy space converges, along a sequence of times and in some weak sense, to a
solution with the compactness property, that is a solution whose trajectory
stays in a compact subset of the energy...

Consider a bounded solution of the focusing, energy-critical wave equation
that does not scatter to a linear solution. We prove that this solution
converges in some weak sense, along a sequence of times and up to scaling and
space translation, to a sum of solitary waves. This result is a consequence of
a new general compactness/rigidity argument ba...

These are lectures notes for a 6 hour course given at PCMI, Park City, Utah,
in the summer of 2003. The notes are based on a series of joint works of
Kenig-Ponce-Vega. Each lecture had problems assigned with it. The lecture notes
were prepared with the help of Justin Holmer. Justin Holmer also provided
solutions to the problems, in an appendix to t...

We consider equivariant solutions for the Schrödinger map problem from ℝ2+1 to S2 with energy less than 4π and show that they are global in time and scatter.

We survey recent results on Calderon's inverse problem with partial data,
focusing on three and higher dimensions.

The present paper establishes a certain duality between the Dirichlet and
Regularity problems for elliptic operators with $t$-independent complex bounded
measurable coefficients ($t$ being the transversal direction to the boundary).
To be precise, we show that the Dirichlet boundary value problem is solvable in
$L^{p'}$, subject to the square funct...

In this paper we study 1-equivariant wave maps of finite energy from
1+3-dimensional Minkowski space exterior to the unit ball at the origin into
the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the
unit sphere in R^3 gets mapped to the north pole. Finite energy implies that
spacial infinity gets mapped to either the nort...

We consider equivariant solutions for the Schr\"odinger map problem from
$\R^{2+1}$ to $\H^2$ with finite energy and show that they are global in time
and scatter.

We consider Calderon's inverse problem with partial data in dimensions $n
\geq 3$. If the inaccessible part of the boundary satisfies a (conformal)
flatness condition in one direction, we show that this problem reduces to the
invertibility of a broken geodesic ray transform. In Euclidean space, sets
satisfying the flatness condition include parts o...

In these lectures I will describe a program (which I will call the concentrationcompactness/rigidity method) that Frank Merle and I have been developing to study critical evolution problems. The issues studied center around global wellposedness and scattering. The method applies to nonlinear dispersive and wave equations in both defocusing and focu...

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere of
finite energy. We establish a classification of all degree 1 global solutions
whose energies are less than three times the energy of the harmonic map Q. In
particular, for each global energy solution of topological degree 1, we show
that the solution asymptotically decouples...

We consider the radial free wave equation in all dimensions and derive
asymptotic formulas for the space partition of the energy relative to a light
cone, as time goes to infinity. We show that the exterior energy estimate,
which Duyckaerts, Merle and the second author obtained in odd dimensions, fails
in even dimensions. Positive results for restr...

For a family of elliptic operators with rapidly oscillating periodic
coefficients, we study the convergence rates for Dirichlet eigenvalues
and bounds of the normal derivatives of Dirichlet eigenfunctions. The
results rely on an $O(\epsilon)$ estimate in $H^1$ for solutions with
Dirichlet condition.

In this paper, we consider the wave equation in space dimension 3 with an
energy-supercritical, focusing nonlinearity. We show that any radial solution
of the equation which is bounded in the critical Sobolev space is globally
defined and scatters to a linear solution. As a consequence, finite time
blow-up solutions have critical Sobolev norm conve...

We prove that if a solution of an equation of KdV type is bounded above by a
traveling wave with an amplitude that decays faster than a given linear
exponential then it must be zero. We assume no restrictions neither on the size
nor in the direction of the speed of the traveling wave.

We prove that the initial value problem (IVP) associated to the fifth order
KdV equation {equation} \label{05KdV} \partial_tu-\alpha\partial^5_x
u=c_1\partial_xu\partial_x^2u+c_2\partial_x(u\partial_x^2u)+c_3\partial_x(u^3),
{equation} where $x \in \mathbb R$, $t \in \mathbb R$, $u=u(x,t)$ is a
real-valued function and $\alpha, \ c_1, \ c_2, \ c_3$...

In this paper, we describe the asymptotic behaviour of globally defined
solutions and of bounded solutions blowing up in finite time of the radial
energy-critical focusing non-linear wave equation in three space dimension.

We consider divergence form elliptic operators L = - div A(x)\nabla, defined
in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is
bounded, measurable, uniformly elliptic, t-independent, and not necessarily
symmetric. We establish square function/non-tangential maximal function
estimates for solutions of the homogeneous equati...

We prove unique continuation principles for solutions of evolution
Schr\"odinger equations with time dependent potentials. These correspond to
uncertainly principles of Paley-Wiener type for the Fourier transform. Our
results extends to a large class of semi-linear Schr\"odinger equation.

In this paper we consider global and non-global bounded radial solutions of
the focusing energy-critical wave equation in space dimension 3. We show that
any of these solutions decouples, along a sequence of times that goes to the
maximal time of existence, as a sum of modulated stationary solutions, a free
radiation term and a term going to 0 in t...

In this note we review recent joint works of the author and F. Merle [Acta Math. 201, No. 2, 147–212 (2008; Zbl 1183.35202); Trans. Am. Math. Soc. 362, No. 4, 1937–1962 (2010; Zbl 1188.35180)] and the author, T. Duyckaerts and F. Merle [J. Eur. Math. Soc. (JEMS) 13, No. 3, 533–599 (2011; Zbl 1230.35067); J. Eur. Math. Soc. (JEMS) 14, No. 5, 1389–14...

In this article we prove $L^p$ estimates for resolvents of Laplace-Beltrami
operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz
and Sogge in the Euclidean case and Shen for the torus. We follow Sogge and
construct Hadamard's parametrix, then use classical boundedness results on
integral operators with oscillatory kernels...

We discuss recent progress in the understanding of the global behavior of solutions to critical non-linear dispersive equations. The emphasis is on global existence, scattering and finite time blow-up. For solutions that are bounded in the critical norm, but which blow-up in finite time, we also discuss the issue of universal profiles at the blow-u...

We consider equivariant solutions for the Schr\"odinger map problem from
$\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ with energy less than $4\pi$ and
show that they are global in time and scatter.

We establish pointwise decay bounds for radial, compact solutions of energy supercritical wave equations in odd dimensions. Applications are given.

Boundary effects play a crucial role in the dynamics of gases governed by the Boltzmann equation

In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and analyticity of variational (“energy
minimizing weak”) solutions to nonlinear elliptic variational problems. In so doing, he developed a very geometric, basic
method to deduce boundedness and regularity of solutions to a priori very discontinuous problems. The es...

In this work we shall review some of our recent results concerning unique
continuation properties of solutions of Schr\"odinger equations. In this
equations we include linear ones with a time depending potential and
semi-linear ones.

We prove an optimal restriction theorem for an arbitrary homogeneous
polynomial hypersurface (of degree at least 2) in R^3, with affine curvature
introduced as mitigating factor.

We consider the Schrödinger map initial-value problem ∂tφ = φ × ∆φ on R d × R; φ(0) = φ0, where φ: R d × R → S 2 ֒ → R 3 is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ∈ C(R: H ∞ Q), Q ∈ S2, provided that the data φ0 ∈ H ∞ Q is smooth and satisfies the...

In these lectures I will describe a program (which I will call the concentrationcompactness/rigidity method) that Frank Merle
and I have been developing to study critical evolution problems. The issues studied center around global wellposedness and
scattering. The method applies to nonlinear dispersive and wave equations in both defocusing and focu...

In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann,
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 poten...

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of
elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients
in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a
consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov
eigenvalues of {L_\epsilon}. Most of our r...

## Citations

... For higher dimensions N ≥ 6 other non-radiative solutions exist, which is proved in the present article. The strategy of [10] could still be adapted to the N = 6 dimensional case by [2], by introducing new weakened channels of energy estimates in the vicinity of a multisoliton like (1.5) but with a logarithmic loss [2], and by classifying the behaviour of non-radiative solutions at infinity, which is the purpose of the present article. We believe this analysis can be extended to k ≥ 2-equivariant wave maps (1.2) as well as the radial 4 + 1 dimensional Yang-Mills system since both the weakened channels of energy estimates of [2] and the classification of non-radiative solutions are valid. ...

... Then u has a finite order of vanishing in B R .0/, which follows from a doubling property proven in [1] and later in [8] using a different method. Moreover, in a previous paper [9], we proved a more precise decay rate for such function (see Lemma 2.4); more importantly, we gave an estimate of the size of the singular set Ã.u/ WD ¹X 2 D \ B R .0/ W u.X / D 0 D jru.X /jº: ...

... Then we use (4.3), (4.10), (4.6), (4.7) to bound this by 5 4 ⟨t⟩] t∇ 2 φ L ∞ ≲ ǫ⟨t⟩ −1 ǫ 2 ǫ⟨t⟩ −1 2 + ǫ⟨t⟩ −1 2 ǫD α ǫ⟨t⟩ −1 2 + ǫ⟨t⟩ −1 2 ǫD α ǫ⟨t⟩ −1 2 + ǫ⟨t⟩ −1 2 ǫ 2 ǫ⟨t⟩ −1 ≲ ǫ 3 ⟨t⟩ −1 D N + ǫ 4 ⟨t⟩ −3 2 ≲ ǫ 2 D 2 N + ǫ 4 ⟨t⟩ −3 2 . Hence, the J 26 is bounded by J 26 ≲ ǫ 2 D 2 N + ǫ 4 ⟨t⟩ −3 2+2δ . ...

... For classic references on the subject we refer the reader to Sunada [48], Atiyah, Donnelly and Singer [1], Borel and Garland [2], Jerison and Lebeau [28], Donnelly and Fefferman [13,14,15,16], Donnelly and Garofalo [17,18], and Lin [36]. As for recent works on the subject we refer the reader to Apraiz, Escauriaza, Wang, and Zhang [3], Blair and Sogge [6], Cavalletti and Farinelli [9], Enciso and Peralta-Salas [21], Georgiev [25], Kenig, Zhu, and Zhuge [29], Logunov [30,31], Logunov, Malinnikova, Nadirashvili, and Nazarov [32], Tian and Yang [50] and Toth and Zelditch [51] just to mention a few. About the applications of spectral inequalities to the control theory we refer to Benabdallah and Naso [4], Fu, Lü, and Zhang [24], Lebeau and Robbiano [34], Lebeau and Zuazua [35], J.-L. ...

... We discuss the classification of all radial weakly non-radiative solutions to (CP1) in this work. Duyckaerts-Kenig-Merle [6] describes the asymptotic behaviours of all radial weakly non-radiative solutions to the focusing equation ∂ 2 t u − ∆u = +|u| 4/(d−2) u in all odd dimensions d ≥ 5. For simplicity we only give the statement in 5-dimensional case here, although higher dimensions are similar. ...

... The past decade has seen a significant progress in mathematical understanding of the soliton resolution, especially for radial solutions of the energy critical nonlinear wave equations (see [2] for a survey and references therein). Notably, the soliton resolution was recently proved for the equivariant wave maps R 2+1 → S 2 [3,4] and the equivariant Yang-Mills (YM) equation in 4 + 1 dimensions [4] (both for the global and blowup solutions). ...

... Under the conditions (1.4), (1.5) and (1.6), we were able to show that H d−1 x ∈ E 1/2 : u ε (x) = 0 ≤ C(N ), (1.13) where C(N ) depends at most on d, λ, Γ, M , and N . The question of explicit dependence of C(N ) on N in a doubling inequality, which plays a key role in the proof of (1. 13), was subsequently addressed in [14,15,1]. This paper continues the study of geometric properties for the operator L ε , arising in the theory of homogenization. ...

... In the recent years, extensive literature was devoted to the use of probabilistic techniques, most often combined with modern Fourier Analysis, in the study of the questions of local well-posedness for rough initial data, as well as global well-posedness for supercritical equations. Among the notable contributions, we mention the breakthrough results of Burq and Tzvetkov [7,8] on supercritical wave equations, the recent theory of random tensors of Deng, Nahmod and Yue [12], global well-posedness results for energycritical equations by Bringmann [2], Kenig and Mendelson [16], Krieger, Lührmann and Staffilani [18]. Other important advances include [13,17,20,9], and we would like to refer the reader to these works for a fuller bibliography. ...

... See the companion paper [1] for further discussion of the impact of Bourgain's "Ribe program". Dyadic pigeonholing makes a small but important role in an important result [9] of Bourgain on the energy-critical nonlinear Schrödinger equation (NLS), discussed in more detail in Kenig's article [20]: At one stage [9, §4] in the (rather intricate) argument, a solution u is constructed to exhibit a concentration property at a certain time t j s and frequency N j s , in that (suppressing a parameter η that is not relevant for the current discussion) ...

Reference: Exploring the toolkit of Jean Bourgain

... + N −2 2 W is in the radial kernel of −∆ + V . For odd dimensions, the strong estimates (1.5) and their extensions to domains {|x| > R + |t|} for R > 0 allowed the authors of [DKM20] to extend this estimate to Equation (1.1) in the radial case: ...