Charles Fefferman

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Charles Louis Fefferman is an American mathematician famous for his work in mathematical analysis. Born on April 18, 1949, he became a professor at Princeton University at a young age. Fefferman has made deep contributions to several areas, including complex analysis, partial differential equations, and Fourier analysis. He received the Fields Medal in 1978, one of the highest honors in mathematics, in recognition of his influence on and advancements in the field of mathematical analysis.

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All published works (247)

Preface for the special issue in honor of Robert Fefferman

2025-05-21

Charles Fefferman , Guozhen Lu

Quantum tunneling and its absence in deep wells and strong magnetic fields

2024-12-30

Charles Fefferman , Jacob Shapiro , Michael I. Weinstein

We present new results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double well potentials containing examples … We present new results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to anti-symmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.

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Sobolev extension in a simple case

2024-05-24

Marjorie Drake , Charles Fefferman , Kevin Ren +1 more

Abstract In this paper, we establish the existence of a bounded, linear extension operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> … Abstract In this paper, we establish the existence of a bounded, linear extension operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{2}$ contained in a line.

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Almost Optimal Agnostic Control of Unknown Linear Dynamics

2024-03-10

Jacob Carruth , Maximilian F. Eggl , Charles Fefferman +1 more

We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control … We consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. We study three variants of the control problem: Bayesian control, in which we have a prior belief about $a$; bounded agnostic control, in which we have no prior belief about $a$ but we assume that $a$ belongs to a bounded set; and fully agnostic control, in which $a$ is allowed to be an arbitrary real number about which we have no prior belief. In the Bayesian variant, a control strategy is optimal if it minimizes a certain expected cost. In the agnostic variants, a control strategy is optimal if it minimizes a quantity called the worst-case regret. For the Bayesian and bounded agnostic variants above, we produce optimal control strategies. For the fully agnostic variant, we produce almost optimal control strategies, i.e., for any $\varepsilon>0$ we produce a strategy that minimizes the worst-case regret to within a multiplicative factor of $(1+\varepsilon)$.

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Sobolev extension in a simple case

2024-02-18

Marjorie Drake , Charles Fefferman , Kevin Ren +1 more

In this paper, we establish the existence of a bounded, linear extension operator $T: L^{2,p}(E) \to L^{2,p}(\mathbb{R}^2)$ when $1<p<2$ and $E$ is a finite subset of $\mathbb{R}^2$ contained in a … In this paper, we establish the existence of a bounded, linear extension operator $T: L^{2,p}(E) \to L^{2,p}(\mathbb{R}^2)$ when $1<p<2$ and $E$ is a finite subset of $\mathbb{R}^2$ contained in a line.

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Discrete honeycombs, rational edges, and edge states

2023-09-22

Charles Fefferman , Sonia Fliss , Michael I. Weinstein

Consider the tight binding model of graphene, sharply terminated along an edge ${\bf l}$ parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges … Consider the tight binding model of graphene, sharply terminated along an edge ${\bf l}$ parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges ${\bf l}$ into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most ${\bf l}$.

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A property of ideals of jets of functions vanishing on a set

2023-04-05

Charles Fefferman , Ary Shaviv

For a set E\subset\mathbb{R}^n that contains the origin, we consider I^m(E) – the set set of all m^{\text{th}} degree Taylor approximations (at the origin) of C^m functions on \mathbb{R}^n that … For a set E\subset\mathbb{R}^n that contains the origin, we consider I^m(E) – the set set of all m^{\text{th}} degree Taylor approximations (at the origin) of C^m functions on \mathbb{R}^n that vanish on E . This set is a proper ideal in \mathcal{P}^m(\mathbb{R}^n) – the ring of all m^{\text{th}} degree Taylor approximations of C^m functions on \mathbb{R}^n . Which ideals in \mathcal{P}^m(\mathbb{R}^n) arise as I^m(E) for some E ? In this paper we introduce the notion of a closed ideal in \mathcal{P}^m(\mathbb{R}^n) , and prove that any ideal of the form I^m(E) is closed. We do not know whether in general any closed proper ideal is of the form I^m(E) for some E , however we prove in a subsequent paper that all closed proper ideals in \mathcal{P}^m(\mathbb{R}^n) arise as I^m(E) when m+n\leq5 .

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Linear extension operators for Sobolev spaces on radially symmetric binary trees

2023-01-01

Charles Fefferman , Bo’az Klartag

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> 1\lt p\lt \infty and suppose that we are given a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f defined on the leaves … Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> 1\lt p\lt \infty and suppose that we are given a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f defined on the leaves of a weighted tree. We would like to extend <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f to a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F defined on the entire tree, so as to minimize the weighted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>W</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:math> {W}^{1,p} -Sobolev norm of the extension. An easy situation is when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> p=2 , where the harmonic extension operator provides such a function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> </m:math> F . In this note, we record our analysis of the particular case of a radially symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p . This operator is a variant of the standard harmonic extension operator, and in fact, it is harmonic extension with respect to a certain Markov kernel determined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p and by the weights.

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Optimal Agnostic Control of Unknown Linear Dynamics in a Bounded Parameter Range

2023-01-01

Jacob Carruth , Maximilian F. Eggl , Charles Fefferman +1 more

Here and in a follow-on paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In … Here and in a follow-on paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, we assume that $a$ is bounded, i.e., that $|a| \le a_{\text{MAX}}$, and we study two variants of the control problem. In the first variant, Bayesian control, we are given a prior probability distribution for $a$ and we seek a strategy that minimizes the expected value of a given cost function. Assuming that we can solve a certain PDE (the Hamilton-Jacobi-Bellman equation), we produce optimal strategies for Bayesian control. In the second variant, agnostic control, we assume nothing about $a$ and we seek a strategy that minimizes a quantity called the regret. We produce a prior probability distribution $d\text{Prior}(a)$ supported on a finite subset of $[-a_{\text{MAX}},a_{\text{MAX}}]$ so that the agnostic control problem reduces to the Bayesian control problem for the prior $d\text{Prior}(a)$.

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Controlling Unknown Linear Dynamics with Almost Optimal Regret

2023-01-01

Jacob Carruth , Maximilian F. Eggl , Charles Fefferman +1 more

Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In … Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, we assume that $a$ can be any real number and we do not assume that we have a prior belief about $a$. We seek a control strategy that minimizes a quantity called the regret. Given any $\varepsilon>0$, we produce a strategy that minimizes the regret to within a multiplicative factor of $(1+\varepsilon)$.

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Fitting a manifold to data in the presence of large noise

2023-01-01

Charles Fefferman , S. V. Ivanov , Matti Lassas +1 more

We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 … We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 \subseteq \partial K_0 \subseteq B^n_1(0).$ We also suppose that $M_0$ has volume ($d$-dimensional Hausdorff measure) less or equal to $V$, reach (i.e., normal injectivity radius) greater or equal to $\tau$. Moreover, we assume that $M_0$ is $R$-exposed, that is, tangent to every point $x \in M$ there is a closed ball of radius $R$ that contains $M$. Let $x_1, \dots, x_N$ be independent random variables sampled from uniform distribution on $M_0$ and $\zeta_1, \dots, \zeta_N$ be a sequence of i.i.d Gaussian random variables in $R^n$ that are independent of $x_1, \dots, x_N$ and have mean zero and covariance $\sigma^2 I_n.$ We assume that we are given the noisy sample points $y_i$, given by $$ y_i = x_i + \zeta_i,\quad \hbox{ for }i = 1, 2, \dots,N. $$ Let $\epsilon,\eta>0$ be real numbers and $k\geq 2$. Given points $y_i$, $i=1,2,\dots,N$, we produce a $C^k$-smooth function which zero set is a manifold $M_{rec}\subseteq R^n$ such that the Hausdorff distance between $M_{rec}$ and $M_0$ is at most $ \epsilon$ and $M_{rec}$ has reach that is bounded below by $c\tau/d^6$ with probability at least $1 - \eta.$ Assuming $d < c \sqrt{\log \log n}$ and all the other parameters are positive constants independent of $n$, the number of the needed arithmetic operations is polynomial in $n$. In the present work, we allow the noise magnitude $\sigma$ to be an arbitrarily large constant, thus overcoming a drawback of previous work.

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Controlling unknown linear dynamics with bounded multiplicative regret

2022-12-22

Jacob Carruth , Maximilian F. Eggl , Charles Fefferman +2 more

We consider a simple control problem in which the underlying dynamics depend on a parameter that is unknown and must be learned. We exhibit a control strategy which is optimal … We consider a simple control problem in which the underlying dynamics depend on a parameter that is unknown and must be learned. We exhibit a control strategy which is optimal to within a multiplicative constant. While most authors find strategies which are successful as the time horizon tends to infinity, our strategy achieves lowest expected cost up to a constant factor for a fixed time horizon.

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Edge states in rationally terminated honeycomb structures

2022-11-15

Charles Fefferman , Sonia Fliss , Michael I. Weinstein

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l … Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type,” generalizing the classical zigzag and armchair edges. We prove that zero-energy/flat-band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l.

Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces

2022-07-10

Charles Fefferman , Karol W. Hajduk , James C. Robinson

We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and … We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by using the explicit form of these fractional power spaces for the Dirichlet Laplacian and Stokes operators. As a simple application, we prove that all weak solutions of the convective Brinkman–Forchheimer equations posed on a bounded domain in R 3 ${\mathbb {R}}^3$ satisfy the energy equality.

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$C^2$ interpolation with range restriction

2022-06-24

Charles Fefferman , Fushuai Jiang , Garving K. Luli

Given -\infty< \lambda < \Lambda < \infty , E \subset \mathbb{R}^n finite, and f\colon E \to [\lambda,\Lambda] , how can we extend f to a C^m(\mathbb{R}^n) function F such that … Given -\infty< \lambda < \Lambda < \infty , E \subset \mathbb{R}^n finite, and f\colon E \to [\lambda,\Lambda] , how can we extend f to a C^m(\mathbb{R}^n) function F such that \lambda\leq F \leq \Lambda and \|F\|_{C^m(\mathbb{R}^n)} is within a constant multiple of the least possible, with the constant depending only on m and n ? In this paper, we provide the solution to the problem for the case {m = 2} . Specifically, we construct a (parameter-dependent, nonlinear) C^2(\mathbb{R}^n) extension operator that preserves the range [\lambda,\Lambda] , and we provide an efficient algorithm to compute such an extension using O(N\log N) operations, where N = \#(E) .

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$C^{m}$ semialgebraic sections over the plane

2022-02-03

Charles Fefferman , Garving K. Luli

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic … In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^2, \mathbb{R}^{D})$. If $\mathcal{H}$ has a section, then it has a semialgebraic section.

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Lower Bound on Quantum Tunneling for Strong Magnetic Fields

2022-02-01

Charles Fefferman , Jacob Shapiro , Michael I. Weinstein

We consider a particle bound to a two-dimensional plane and a double-well potential, subject to a perpendicular uniform magnetic field. The energy difference between the lowest two eigenvalues---the eigenvalue splitting---is … We consider a particle bound to a two-dimensional plane and a double-well potential, subject to a perpendicular uniform magnetic field. The energy difference between the lowest two eigenvalues---the eigenvalue splitting---is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping amplitude between the wells, a key parameter in tight binding models of solid state physics, given by an oscillatory integral, whose phase has no critical point and which is exponentially small.

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Optimal Control with Learning on the Fly: System with Unknown Drift

2022-01-01

Daniel Gurevich , Debdipta Goswami , Charles Fefferman +1 more

This paper derives an optimal control strategy for a simple stochastic dynamical system with constant drift and an additive control input. Motivated by the example of a physical system with … This paper derives an optimal control strategy for a simple stochastic dynamical system with constant drift and an additive control input. Motivated by the example of a physical system with an unexpected change in its dynamics, we take the drift parameter to be unknown, so that it must be learned while controlling the system. The state of the system is observed through a linear observation model with Gaussian noise. In contrast to most previous work, which focuses on a controller's asymptotic performance over an infinite time horizon, we minimize a quadratic cost function over a finite time horizon. The performance of our control strategy is quantified by comparing its cost with the cost incurred by an optimal controller that has full knowledge of the parameters. This approach gives rise to several notions of "regret." We derive a set of control strategies that provably minimize the worst-case regret; these arise from Bayesian strategies that assume a specific fixed prior on the drift parameter. This work suggests that examining Bayesian strategies may lead to optimal or near-optimal control strategies for a much larger class of realistic dynamical models with unknown parameters.

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A property of ideals of jets of functions vanishing on a set

2022-01-01

Charles Fefferman , Ary Shaviv

For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish … For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is an ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. Which ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ for some $E$? In this paper we introduce the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and prove that any ideal of the form $I^m(E)$ is closed. We do not know whether in general any closed ideal is of the form $I^m(E)$ for some $E$, however we prove in [FS] that all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$.

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Classification of implication-closed ideals in certain rings of jets

2022-01-01

Charles Fefferman , Ary Shaviv

For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish … For a set $E\subset\mathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{\text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $\mathbb{R}^n$ that vanish on $E$. This set is a proper ideal in $\mathcal{P}^m(\mathbb{R}^n)$ -- the ring of all $m^{\text{th}}$ degree Taylor approximations of $C^m$ functions on $\mathbb{R}^n$. In [FS] we introduced the notion of a \textit{closed} ideal in $\mathcal{P}^m(\mathbb{R}^n)$, and proved that any ideal of the form $I^m(E)$ is closed. In this paper we classify (up to a natural equivalence relation) all closed ideals in $\mathcal{P}^m(\mathbb{R}^n)$ in all cases in which $m+n\leq5$. We also show that in these cases the converse also holds -- all closed proper ideals in $\mathcal{P}^m(\mathbb{R}^n)$ arise as $I^m(E)$ when $m+n\leq5$. In addition, we prove that in these cases any ideal of the form $I^m(E)$ for some $E\subset\mathbb{R}^n$ that contains the origin already arises as $I^m(V)$ for some semi-algebraic $V\subset\mathbb{R}^n$ that contains the origin. By doing so we prove that a conjecture by N. Zobin holds true in these cases.

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Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

2021-09-27

Charles Fefferman , Benjamin C. Pooley , José L. Rodrigo

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Invariant Curves for Degenerate Hyperbolic Maps of the Plane

2021-08-10

Charles Fefferman

We prove existence and uniqueness of an unstable manifold for a degenerate hyperbolic map of the plane arising in statistics. We prove existence and uniqueness of an unstable manifold for a degenerate hyperbolic map of the plane arising in statistics.

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Optimal control with learning on the fly: a toy problem

2021-06-21

Charles Fefferman , Bernat Guillén Pegueroles , Clarence W. Rowley +1 more

We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a … We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and “agnostic” control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of “regret”, and we obtain strategies that minimize the “worst-case regret” arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

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A Bounded mean oscillation (BMO) theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$ and PDE

2021-01-01

Charles Fefferman , Steven B. Damelin

This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in … This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in the memoir preprint: arXiv:2103.09748, [1], submitted for consideration for publication.

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Invariant Curves for Degenerate Hyperbolic Maps of the Plane

2021-01-01

Charles Fefferman

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$C^m$ Semialgebraic Sections Over the Plane

2021-01-01

Charles Fefferman , Garving K. Luli

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic … In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C^m_{loc}\left( \mathbb{R}^{2},\mathbb{R}^{D}\right) .$ If $\mathcal{H}$ has a section, then it has a semialgebraic section.

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The Classical Whitney Extension Problem

2020-10-28

Charles Fefferman , Arie Israel

Extension and Interpolation in Sobolev Spaces

2020-10-28

Charles Fefferman , Arie Israel

Vector-Valued Functions

2020-10-28

Charles Fefferman , Arie Israel

Continuum Schroedinger Operators for Sharply Terminated Graphene-Like Structures

2020-10-21

Charles Fefferman , Michael I. Weinstein

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Generators for the $C^m$-closures of ideals

2020-06-30

Charles Fefferman , Garving K. Luli

Let \mathscr{R} denote the ring of real polynomials on \mathbb{R}^{n} . Fix m\geq 0 , and let A_{1},\ldots,A_{M}\in\mathscr{R} . The C^{m} -closure of (A_{1},\ldots,A_{M}) , denoted here by [A_{1},\ldots,A_{M};C^{m}] , … Let \mathscr{R} denote the ring of real polynomials on \mathbb{R}^{n} . Fix m\geq 0 , and let A_{1},\ldots,A_{M}\in\mathscr{R} . The C^{m} -closure of (A_{1},\ldots,A_{M}) , denoted here by [A_{1},\ldots,A_{M};C^{m}] , is the ideal of all f\in \mathscr{R} expressible in the form f=F_{1}A_{1}+\cdots +F_{M}A_{M} with each F_{i}\in C^{m}(\mathbb{R}^{n}) . In this paper we exhibit an algorithm for computing generators for [A_{1},\ldots,A_{M};C^{m}] .

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Solutions to a system of equations for $C^m$ functions

2020-06-30

Charles Fefferman , Garving K. Luli

Fix m\geq 0 , and let A=(A_{ij}(x))_{1 \leq i \leq N, 1\leq j \leq M} be a matrix of semialgebraic functions on \mathbb{R}^n or on a compact subset E \subset … Fix m\geq 0 , and let A=(A_{ij}(x))_{1 \leq i \leq N, 1\leq j \leq M} be a matrix of semialgebraic functions on \mathbb{R}^n or on a compact subset E \subset \mathbb{R}^n . Given f=(f_1,\ldots,f_N) \in C^\infty(\mathbb{R}^n, \mathbb{R}^N) , we consider the following system of equations: \sum_{j=1}^M A_{ij} (x) F_j (x) = f_i (x) \quad\text{for } i =1,\ldots, N. In this paper, we give algorithms for computing a finite list of linear partial differential operators such that AF=f admits a C^m(\mathbb{R}^n,\mathbb{R}^M) solution F=(F_1,\ldots,F_M) if and only if f=(f_1,\ldots,f_N) is annihilated by the linear partial differential operators.

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Defect Modes for Dislocated Periodic Media

2020-06-19

Alexis Drouot , Charles Fefferman , Michael I. Weinstein

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Analysis and applications: The mathematical work of Elias Stein

2020-03-03

Charles Fefferman , Alex Ionescu , Terence Tao +1 more

This article discusses some of Elias M. Stein's seminal contributions to analysis. This article discusses some of Elias M. Stein's seminal contributions to analysis.

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Optimal control with learning on the fly: a toy problem

2020-02-26

Charles Fefferman , Bernat Guillén Pegueroles , Clarence W. Rowley +1 more

We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a … We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of regret, and we obtain strategies that minimize the worst-case regret arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

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Lower Bound on Quantum Tunneling for Strong Magnetic Fields

2020-01-01

Charles Fefferman , Jacob N. Shapiro , Michael I. Weinstein

We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the … We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest two eigenvalues--the eigenvalue splitting--is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping probability between the wells, a key parameter in tight binding models of solid state physics.

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Optimal control with learning on the fly: a toy problem

2020-01-01

Charles Fefferman , Bernat Guillén Pegueroles , Clarence W. Rowley +1 more

We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a … We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned. We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and "agnostic" control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of "regret," and we obtain strategies that minimize the "worst-case regret" arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

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Efficient Algorithms for Approximate Smooth Selection

2019-08-13

Charles Fefferman , Bernat Guillén Pegueroles

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Splash Singularities for the Free Boundary Navier-Stokes Equations

2019-05-11

Ángel Castro , Diego Córdoba , Charles Fefferman +2 more

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Efficient Algorithms for Approximate Smooth Selection

2019-05-08

Charles Fefferman , Bernat Guillén Pegueroles

In this paper we provide efficient algorithms for approximate $\mathcal{C}^m(\mathbb{R}^n, \mathbb{R}^D)-$selection. In particular, given a set $E$, constants $M_0 > 0$ and $0 <\tau \leq \tau_{\max}$, and convex sets $K(x) … In this paper we provide efficient algorithms for approximate $\mathcal{C}^m(\mathbb{R}^n, \mathbb{R}^D)-$selection. In particular, given a set $E$, constants $M_0 > 0$ and $0 <\tau \leq \tau_{\max}$, and convex sets $K(x) \subset \mathbb{R}^D$ for $x \in E$, we show that an algorithm running in $C(\tau) N \log N$ steps is able to solve the smooth selection problem of selecting a point $y \in (1+\tau)\blacklozenge K(x)$ for $x \in E$ for an appropriate dilation of $K(x)$, $(1+\tau)\blacklozenge K(x)$, and guaranteeing that a function interpolating the points $(x, y)$ will be $\mathcal{C}^m(\mathbb{R}^n, \mathbb{R}^D)$ with norm bounded by $C M_0$.

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Generators for the $C^m$-closures of Ideals

2019-02-11

Charles Fefferman , Garving K. Luli

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ … Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $. In this paper we exhibit an algorithm to compute generators for $\left[ A_{1},\cdots ,A_{M};C^{m}\right] $.

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Solutions to A System of Equations for $C^m$ Functions

2019-02-11

Charles Fefferman , Garving K. Luli

Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. … Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=\left( f_{1},\cdots ,f_{N}\right) \in C^{\infty }\left( \mathbb{R}^{n},\mathbb{R}^{N}\right) $, we consider the following system of equations \begin{equation} \sum_{j=1}^{M}A_{ij}\left( x\right) F_{j}\left( x\right) =f_{i}\left( x\right) \text{ }\left( i=1,\cdots ,N\right) \text{.} \end{equation} In this paper, we give algorithms for computing a finite list of linear partial differential operators such that $AF= f$ admits a $C^m(\mathbb{R}^n, \mathbb{R}^M)$ solution $F=(F_1,\cdots, F_M)$ if and only if $f=(f_1,\cdots, f_N)$ is annihilated by the linear partial differential operators.

Efficient Algorithms for Approximate Smooth Selection

2019-01-01

Charles Fefferman , Bernat Guillén Pegueroles

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Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces

2019-01-01

Charles Fefferman , Karol W. Hajduk , James C. Robinson

We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and … We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by using the explicit form of these fractional power spaces for the Dirichlet Laplacian and Stokes operators. As a simple application, we prove that all weak solutions of the incompressible convective Brinkman--Forchheimer equations posed on a bounded domain in ${\mathbb R}^3$ satisfy the energy equality.

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Generators for the $C^m$-closures of Ideals

2019-01-01

Charles Fefferman , Garving K. Luli

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Defect modes for dislocated periodic media

2018-10-13

Alexis Drouot , Charles Fefferman , Michael I. Weinstein

We study defect modes in a one-dimensional periodic medium with a dislocation. The model is a periodic Schrodinger operator on $\mathbb{R}$, perturbed by an adiabatic dislocation of amplitude $\delta\ll 1$. … We study defect modes in a one-dimensional periodic medium with a dislocation. The model is a periodic Schrodinger operator on $\mathbb{R}$, perturbed by an adiabatic dislocation of amplitude $\delta\ll 1$. If the periodic background admits a Dirac point $-$ a linear crossing of dispersion curves $-$ then the dislocated operator acquires a gap in its essential spectrum. For this model (and its 2-dimensional honeycomb analog) Fefferman, Lee-Thorp and Weinstein constructed in previous work defect modes with energies within the gap. The bifurcation of defect modes is associated with the discrete eigenmodes of an effective Dirac operator. We improve upon this result: we show that all the defect modes of the dislocated operator arise from the eigenmodes of the Dirac operator. As a byproduct, we derive full expansions of the eigenpairs in powers of $\delta$. The self-contained proof relies on (a) resolvent estimates for the bulk operators; (b) scattering theory for highly oscillatory potentials developed by the first author. This work significantly advances the understanding of the topological stability of certain defect states, particularly the bulk-edge correspondence for continuous dislocated systems.

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Edge States of continuum Schroedinger operators for sharply terminated honeycomb structures

2018-10-08

Charles Fefferman , Michael I. Weinstein

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Weak-Strong Uniqueness in Fluid Dynamics

2018-09-27

Charles Fefferman , James C. Robinson , José L. Rodrigo

We give a survey of recent results on weak-strong uniqueness for compressible and incompressible Euler and Navier-Stokes equations, and also make some new observations. The importance of the weak-strong uniqueness … We give a survey of recent results on weak-strong uniqueness for compressible and incompressible Euler and Navier-Stokes equations, and also make some new observations. The importance of the weak-strong uniqueness principle stems, on the one hand, from the instances of nonuniqueness for the Euler equations exhibited in the past years; and on the other hand from the question of convergence of singular limits, for which weak-strong uniqueness represents an elegant tool.

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Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”

2018-09-27

Charles Fefferman , James C. Robinson , José L. Rodrigo

This article offers a modern perspective that exposes the many contributions of Leray in his celebrated work on the three-dimensional incompressible Navier-Stokes equations from 1934. Although the importance of his … This article offers a modern perspective that exposes the many contributions of Leray in his celebrated work on the three-dimensional incompressible Navier-Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth or belongs to $$H^1$$ or $$L^2 \cap L^p$$ (with $$p \in (3,\infty]$$ ), as well as lower bounds on the norms $$\| \nabla u (t) \|_2$$ and $$\| u(t) \|_p$$ ( $$p\in(3,\infty]$$ ) as t approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most 1/2. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.

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Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations

2018-09-27

Charles Fefferman , James C. Robinson , José L. Rodrigo

This contribution covers the topic of my talk at the 2016-17 Warwick-EPSRC Symposium: 'PDEs and their applications'. As such it contains some already classical material and some new observations. The … This contribution covers the topic of my talk at the 2016-17 Warwick-EPSRC Symposium: 'PDEs and their applications'. As such it contains some already classical material and some new observations. The main purpose is to compare several avatars of the Kato criterion for the convergence of a Navier-Stokes solution, to a regular solution of the Euler equations, with numerical or physical issues like the presence (or absence) of anomalous energy dissipation, the Kolmogorov 1/3 law or the Onsager C^{0,1/3} conjecture. Comparison with results obtained after September 2016 and an extended list of references have also been added.

Common Coauthors

Coauthor	Papers Together
Michael I. Weinstein	29
Diego Córdoba	27
Garving K. Luli	22
Ángel Castro	16
Francisco Gancedo	16
José L. Rodrigo	15
Arie Israel	14
C. Robin Graham	13
James Lee-Thorp	11
Steven B. Damelin	11
James C. Robinson	10
Javier Gómez-Serrano	9
Clarence W. Rowley	8
Luis Seco	7
Bernat Guillén Pegueroles	6
D. H. Phong	5
Harold Donnelly	5
Melanie Weber	4
Jacob Carruth	4
Maximilian F. Eggl	4
Raghavan Narasimhan	4
Bo′az Klartag	3
J. J. Kohn	3
Gian Michele Graf	3
Alexis Drouot	3
Antonio Cordóba	3
María López-Férnández	3
William Glover	3
Jürg Fröhlich	3
C. Robin Graham	3
Ary Shaviv	3
Harold S. Shapiro	2
Michael Ćwikel	2
Marjorie Drake	2
Pavel Shvartsman	2
Karol W. Hajduk	2
Kevin Ren	2
Anna Skorobogatova	2
Hariharan Narayanan	2
I. Vukićević	2
Peter Constantin	2
Benjamin Muckenhoupt	2
Jacob Shapiro	2
Robert L. Grossman	2
Stephen Wainger	2
Chee Wei Wong	2
Max Jodeit	2
David S. McCormick	2
Michael Beals	2
E. M. Stein	2

Commonly Cited References

The Whitney problem of existence of a linear extension operator

1997-12-01

Yuri Brudnyi , Pavel Shvartsman

A sharp form of Whitney’s extension theorem

2005-01-01

Charles Fefferman

3. Order relations involving multi-indices 4. Statement of two main lemmas 5. Plan of the proof 6. Starting the main induction 7. Nonmonotonic sets 8. A consequence of the main … 3. Order relations involving multi-indices 4. Statement of two main lemmas 5. Plan of the proof 6. Starting the main induction 7. Nonmonotonic sets 8. A consequence of the main inductive assumption 9. Setup for the main induction 10. Applying Helly's theorem on convex sets 11. A Calder6n-Zygmund decomposition 12. Controlling auxiliary polynomials I 13. Controlling auxiliary polynomials II 14. Controlling the main polynomials 15. Proof of Lemmas 9.1 and 5.2 16. A rescaling lemma 17. Proof of Lemma 5.3 18. Proofs of the theorems 19. A bound for k# References

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Differentiable functions defined in closed sets. A problem of Whitney

2003-02-01

Edward Bierstone , Pierre D. Milman , Wiesław Pawłucki

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Whitney's Problem on Extendability of Functions and an Intrinsic Metric

1998-01-01

Nahum Zobin

We consider the space of functions with bounded (k+1)th derivatives in a general domain in Rn. Is every such function extendible to a function of the same class defined on … We consider the space of functions with bounded (k+1)th derivatives in a general domain in Rn. Is every such function extendible to a function of the same class defined on the whole Rn? H. Whitney showed that the equivalence of the intrinsic (=geodesic) metric in this domain to the Euclidean one is sufficient for such extendability. There was an old conjecture (going back to H. Whitney) that this equivalence is also necessary for extendability. We disprove this conjecture and construct examples of domains in R2such that the above extendability holds but the analogous property for smallerkfails. Our study is based on a duality approach.

Interpolation and extrapolation of smooth functions by linear operators

2005-04-30

Charles Fefferman

Let C^{m , 1} ( \mathbb{R}^n) be the space of functions on \mathbb{R}^n whose m^{\sf th} derivatives are Lipschitz 1. For E \subset \mathbb{R}^n , let C^{m , 1} (E) … Let C^{m , 1} ( \mathbb{R}^n) be the space of functions on \mathbb{R}^n whose m^{\sf th} derivatives are Lipschitz 1. For E \subset \mathbb{R}^n , let C^{m , 1} (E) be the space of all restrictions to E of functions in C^{m,1} ( \mathbb{R}^n) . We show that there exists a bounded linear operator T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n) such that, for any f \in C^{m , 1} ( E ) , we have T f = f on E .

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Extension of smooth functions from finitely connected planar domains

1999-09-01

Nahum Zobin

Higher-order tangents and Fefferman’s paper on Whitney’s extension problem

2006-07-01

Edward Bierstone , Pierre D. Milman , Wiesław Pawłucki

Whitney [W2] proved that a function defined on a closed subset of R is the restriction of a C m function if the limiting values of all m th divided … Whitney [W2] proved that a function defined on a closed subset of R is the restriction of a C m function if the limiting values of all m th divided differences form a continuous function.We show that Fefferman's solution of Whitney's problem for R n [F, Th. 1] is equivalent to a variant of our conjecture in [BMP2] giving a criterion for C m extension in terms of iterated limits of finite differences.

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A Generalized Sharp Whitney Theorem for Jets

2005-08-31

Charles Fefferman

Suppose that, for each point x in a given subset E \subset \mathbb{R}^n , we are given an m -jet f(x) and a convex, symmetric set \sigma(x) of m -jets … Suppose that, for each point x in a given subset E \subset \mathbb{R}^n , we are given an m -jet f(x) and a convex, symmetric set \sigma(x) of m -jets at x . We ask whether there exist a function F \in C^{m , \omega} ( \mathbb{R}^n ) and a finite constant M , such that the m -jet of F at x belongs to f ( x ) + M \sigma ( x ) for all x \in E . We give a necessary and sufficient condition for the existence of such F , M , provided each \sigma(x) satisfies a condition that we call "Whitney \omega -convexity''.

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The electronic properties of graphene

2009-01-14

A. H. Castro Neto , F. Guinea , N. M. R. Peres +2 more

This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external … This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

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Analytic Extensions of Differentiable Functions Defined in Closed Sets

1992-01-01

Hassler Whitney

Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry

2008-01-10

F. D. M. Haldane , S. Raghu

We show how, in principle, to construct analogs of quantum Hall edge states in "photonic crystals" made with nonreciprocal (Faraday-effect) media. These form "one-way waveguides" that allow electromagnetic energy to … We show how, in principle, to construct analogs of quantum Hall edge states in "photonic crystals" made with nonreciprocal (Faraday-effect) media. These form "one-way waveguides" that allow electromagnetic energy to flow in one direction only.

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Functions Differentiable on the Boundaries of Regions

1934-07-01

Hassler Whitney

Honeycomb lattice potentials and Dirac points

2012-06-25

Charles Fefferman , Michael I. Weinstein

We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (<italic>Dirac points</italic>) at the vertices of its … We prove that the two-dimensional Schrödinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (<italic>Dirac points</italic>) at the vertices of its Brillouin zone. No assumptions are made on the size of the potential. We then prove the robustness of such conical singularities to a restrictive class of perturbations, which break the honeycomb lattice symmetry. General small perturbations of potentials with Dirac points do not have Dirac points; their dispersion surfaces are smooth. The presence of Dirac points in honeycomb structures is associated with many novel electronic and optical properties of materials such as graphene.

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Functions Differentiable on the Boundaries of Regions

1992-01-01

Hassler Whitney

Differentiable Functions Defined in Closed Sets. I

1992-01-01

Hassler Whitney

Whitney’s extension problem for Cm

2006-07-01

Charles Fefferman

Let f be a real-valued function on a compact set in R n , and let m be a positive integer.We show how to decide whether f extends to aWe … Let f be a real-valued function on a compact set in R n , and let m be a positive integer.We show how to decide whether f extends to aWe answer also the following refinement of Question 1.Question 2. Let ϕ and E be as in Question 1. Fix x ∈ E and P ∈ R x.How can we tell whether there exists F ∈ C m (R n ) with F = ϕ on E and J x(F ) = P ?In particular, we ask which m-jets at x can arise as the jet of a C m function vanishing on E. This is equivalent to determining the "Zariski paratangent space" from Bierstone-Milman-Paw lucki [BMP1].

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Photonic Floquet topological insulators

2013-04-01

Mikael C. Rechtsman , Julia M. Zeuner , Yonatan Plotnik +6 more

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Cmextension by linear operators

2007-11-01

Charles Fefferman

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Lipschitz selections of set-valued mappings and Helly’s theorem

2002-06-01

Pavel Shvartsman

Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions

2001-02-07

Yuri Brudnyi , Pavel Shvartsman

We prove that the trace of the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 comma omega Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> … We prove that the trace of the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 comma omega Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C^{1,\omega }({\mathbb R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to an arbitrary closed subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X\subset {\mathbb R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is characterized by the following “finiteness” property. A function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper X right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f:X\rightarrow {\mathbb R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to the trace space if and only if the restriction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f vertical-bar Subscript upper Y Baseline"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>Y</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">f|_Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to an arbitrary subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y subset-of upper X"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y\subset X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 dot 2 Superscript n minus 1"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">3\cdot 2^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be extended to a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper Y Baseline element-of upper C Superscript 1 comma omega Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_Y\in C^{1,\omega }({\mathbb R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace double-vertical-bar f Subscript upper Y Baseline double-vertical-bar Subscript upper C Sub Superscript 1 comma omega Subscript Baseline colon upper Y subset-of upper X comma c a r d upper Y less-than-or-equal-to 3 dot 2 Superscript n minus 1 Baseline right-brace greater-than normal infinity period"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">sup</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>Y</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>card</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sup \{\|f_Y\|_{C^{1,\omega }}:~Y\subset X, ~\operatorname {card} Y\le 3\cdot 2^{n-1}\}>\infty .</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> The constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 dot 2 Superscript n minus 1"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">3\cdot 2^{n-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.

On an extension theorem

1959-01-01

Hidegorô Nakano

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Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal

2008-01-10

Zheng Wang , Y. D. Chong , John D. Joannopoulos +1 more

We point out that electromagnetic one-way edge modes analogous to quantum Hall edge states, originally predicted by Raghu and Haldane in 2D photonic crystals possessing Dirac point-derived band gaps, can … We point out that electromagnetic one-way edge modes analogous to quantum Hall edge states, originally predicted by Raghu and Haldane in 2D photonic crystals possessing Dirac point-derived band gaps, can appear in more general settings. We show that the TM modes in a gyromagnetic photonic crystal can be formally mapped to electronic wave functions in a periodic electromagnetic field, so that the only requirement for the existence of one-way edge modes is that the Chern number for all bands below a gap is nonzero. In a square-lattice yttrium-iron-garnet crystal operating at microwave frequencies, which lacks Dirac points, time-reversal breaking is strong enough that the effect should be easily observable. For realistic material parameters, the edge modes occupy a 10% band gap. Numerical simulations of a one-way waveguide incorporating this crystal show 100% transmission across strong defects.

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Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

1994-11-01

Peter Constantin , Andrew J. Majda , Esteban G. Tabak

The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents … The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. The possibility of frontogenesis within this approximation is an important issue in the context of geophysical flows. A striking mathematical and physical analogy is developed between the structure and formation of singular solutions of this quasi-geostrophic active scalar in two dimensions and the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric configurations to demonstrate the important role of nontrivial topology in the formation of singular solutions.

Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

2012-01-31

Ángel Castro , Diego Córdoba , Charles Fefferman +2 more

The Muskat problem models the evolution of the interface between two different fluids in porous media.The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem.We show that … The Muskat problem models the evolution of the interface between two different fluids in porous media.The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem.We show that the Rayleigh-Taylor condition may hold initially but break down in finite time.As a consequence of the method used, we prove the existence of water waves turning.

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Finiteness Principles for Smooth Selection

2016-04-01

Charles Fefferman , Arie Israel , Garving K. Luli

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Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

2013-10-11

Gian Michele Graf , Marcello Porta

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Finite time singularities for the free boundary incompressible Euler equations

2013-08-02

Ángel Castro , Diego Córdoba , Charles Fefferman +2 more

In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) … In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.

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Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

2009-11-12

Erwan Le Gruyer

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

2009-12-01

V. V. Grushin

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

1984-03-01

J. Thomas Beale , T. Kato , Andrew J. Majda

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Splash singularity for water waves

2012-01-04

Ángel Castro , Diego Córdoba , Charles Fefferman +2 more

We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability … We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity (self intersecting curve in one point) in finite time.

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Topologically Protected States in One-Dimensional Systems

2017-02-01

Charles Fefferman , James Lee-Thorp , Michael I. Weinstein

We study a class of periodic Schrodinger operators, which we prove have Dirac points. We then show that the introduction of an via adiabatic modulation of a periodic potential by … We study a class of periodic Schrodinger operators, which we prove have Dirac points. We then show that the introduction of an via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized \edge states, associated with the topologically protected zero- energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. Our model also captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene.

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On the Spectra of Carbon Nano-Structures

2007-08-14

Peter Kuchment , Olaf Post

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Zak phase and the existence of edge states in graphene

2011-11-23

Pierre Delplace , Denis Ullmo , Gilles Montambaux

We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the … We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase $\mathcal{Z}({k}_{\ensuremath{\parallel}})$, which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum ${k}_{\ensuremath{\parallel}}$ at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one-dimensional toy model as well as for graphene ribbons with zigzag edges. The range of ${k}_{\ensuremath{\parallel}}$ for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, and it confirms and generalizes the results of several previous works.

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Étude de Quelques Algèbres Tayloriennes

1958-12-01

Georges Glaeser

Well-posedness in Sobolev spaces of the full water wave problem in 2-D

1997-09-03

Sijue Wu

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Whitney’s extension problems and interpolation of data

2008-11-24

Charles Fefferman

Given a function $f: E \rightarrow {\mathbb {R}}$ with $E \subset {\mathbb {R}}^n$, we explain how to decide whether $f$ extends to a $C^m$ function $F$ on ${\mathbb {R}}^n$. If … Given a function $f: E \rightarrow {\mathbb {R}}$ with $E \subset {\mathbb {R}}^n$, we explain how to decide whether $f$ extends to a $C^m$ function $F$ on ${\mathbb {R}}^n$. If $E$ is finite, then one can efficiently compute an $F$ as above, whose $C^m$ norm has the least possible order of magnitude (joint work with B. Klartag).

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Interface evolution: Water waves in 2-D

2009-08-29

Antonio Córdoba , Diego Córdoba , Francisco Gancedo

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Sobolev extension by linear operators

2013-02-28

Charles Fefferman , Arie Israel , Garving K. Luli

Let $L^{m,p}(\mathbb {R}^n)$ be the Sobolev space of functions with $m^{\mathrm {th}}$ derivatives lying in $L^p(\mathbb {R}^n)$. Assume that $n< p < \infty$. For $E \subset \mathbb {R}^n$, let $L^{m,p}(E)$ … Let $L^{m,p}(\mathbb {R}^n)$ be the Sobolev space of functions with $m^{\mathrm {th}}$ derivatives lying in $L^p(\mathbb {R}^n)$. Assume that $n< p < \infty$. For $E \subset \mathbb {R}^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\mathbb {R}^n)$. We show that there exists a bounded linear map $T : L^{m,p}(E) \rightarrow L^{m,p}(\mathbb {R}^n)$ such that, for any $f \in L^{m,p}(E)$, we have $Tf = f$ on $E$. We also give a formula for the order of magnitude of $\|f\|_{L^{m,p}(E)}$ for a given $f : E \rightarrow \mathbb {R}$ when $E$ is finite.

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Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

2016-03-21

Charles Fefferman , James Lee-Thorp , Michael I. Weinstein

Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or 'edge' and are localized transverse to it. This paper summarizes and extends the … Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or 'edge' and are localized transverse to it. This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous two-dimensional (2D) honeycomb structures. We consider a family of Schrödinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous 2D honeycomb structures (http://arxiv.org/abs/1506.06111). The topologically protected edge state bifurcation is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schrödinger operator. Numerical simulations for honeycomb structures of varying contrasts and 'rational edges' (zigzag, armchair and others), support the following scenario: (a) for low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived edge quasi-modes which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge.

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Hypoelliptic differential operators and nilpotent groups

1976-01-01

Linda Preiss Rothschild , E. M. Stein

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Well-posedness in Sobolev spaces of the full water wave problem in 3-D

1999-01-01

Sijue Wu

We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion … We consider the motion of the interface of a 3-D inviscid, incompressible, irrotational water wave, with air region above water region and surface tension zero. We prove that the motion of the interface of the water wave is not subject to Taylor instability, as long as the interface separates the whole 3-D space into two simply connected <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> regions. We prove further the existence and uniqueness of solutions of the full 3-D water wave problem, locally in time, for any initial interface that separates the whole 3-D space into two simply connected regions.

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Colloquium: Topological insulators

2010-11-08

M. Zahid Hasan , C. L. Kane

Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator, but have protected conducting states on their edge or surface. The 2D topological insulator is … Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator, but have protected conducting states on their edge or surface. The 2D topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A 3D topological insulator supports novel spin polarized 2D Dirac fermions on its surface. In this Colloquium article we will review the theoretical foundation for these electronic states and describe recent experiments in which their signatures have been observed. We will describe transport experiments on HgCdTe quantum wells that demonstrate the existence of the edge states predicted for the quantum spin Hall insulator. We will then discuss experiments on Bi_{1-x}Sb_x, Bi_2 Se_3, Bi_2 Te_3 and Sb_2 Te_3 that establish these materials as 3D topological insulators and directly probe the topology of their surface states. We will then describe exotic states that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions, and may provide a new venue for realizing proposals for topological quantum computation. We will close by discussing prospects for observing these exotic states, a well as other potential device applications of topological insulators.

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Evidence of singularities for a family of contour dynamics equations

2005-04-18

Diego Córdoba , Marco A. Fontelos , Ana M. Mancho +1 more

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < alpha </= … In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < alpha </= 1. The limiting case alpha --> 0 corresponds to 2D Euler equations, and alpha = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

Convergence of a Boundary Integral Method for Water Waves

1996-10-01

J. Thomas Beale , Thomas Y. Hou , John Lowengrub

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are … We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration.

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Nonexistence of Simple Hyperbolic Blow-Up for the Quasi-Geostrophic Equation

1998-11-01

Diego Córdoba

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is in the case of Euler … The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak [7] suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible. Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler. These results were announced in [9], but with a slight difference in the definition of a simple hyperbolic saddle. The definition given in Section 4 generalizes the one given in the announcement. See also Constantin [4], discussed in Section 7, Remark 5 below. The whole work described in this paper is basically part of the author's thesis. I am particularly grateful to my thesis advisor Charles Fefferman for his attention, support, guidance and advice. I am indebted to D. Christodoulou and P. Constantin for helpful corrections and suggestions. I wish to thank A. Majda for suggesting the subject and E. Tabak for discussions and com-

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The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

2008-04-09

Pavel Shvartsman

We study a variant of the Whitney extension problem (1934) for the space ${C^{k,\omega }(\mathbf {R}^{n})}$. We identify ${C^{k,\omega }(\mathbf {R}^{n})}$ with a space of Lipschitz mappings from $\mathbf {R}^n$ … We study a variant of the Whitney extension problem (1934) for the space ${C^{k,\omega }(\mathbf {R}^{n})}$. We identify ${C^{k,\omega }(\mathbf {R}^{n})}$ with a space of Lipschitz mappings from $\mathbf {R}^n$ into the space $\mathcal {P}_k\times \mathbf {R}^n$ of polynomial fields on $\mathbf {R}^n$ equipped with a certain metric. This identification allows us to reformulate the Whitney problem for ${C^{k,\omega } (\mathbf {R}^{n})}$ as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of $\mathcal {P}_k\times \mathbf {R}^n$. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for ${C^{k,\omega }(\mathbf {R}^{n})}$ due to C. Fefferman.

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On the evolution of sharp fronts for the quasi‐geostrophic equation

2004-10-21

José L. Rodrigo

Abstract We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two‐dimensional … Abstract We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two‐dimensional Euler equation. The special interest of the quasi‐geostrophic equation lies in its strong similarities with the three‐dimensional Euler equation, while being a two‐dimen‐sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three‐dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot‐Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash‐Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C ∞ case. © 2004 Wiley Periodicals, Inc.

Photonic realization of topologically protected bound states in domain-wall waveguide arrays

2016-03-10

James Lee-Thorp , I. Vukićević , Xiaofang Xu +4 more

We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further … We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further numerically confirm the applicability of this theory for three-dimensional structures.

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Analogs of quantum-Hall-effect edge states in photonic crystals

2008-09-23

S. Raghu , F. D. M. Haldane

Photonic crystals built with time-reversal-symmetry-breaking Faraday-effect media can exhibit chiral edge modes that propagate unidirectionally along boundaries across which the Faraday axis reverses. These modes are precise analogs of the … Photonic crystals built with time-reversal-symmetry-breaking Faraday-effect media can exhibit chiral edge modes that propagate unidirectionally along boundaries across which the Faraday axis reverses. These modes are precise analogs of the electronic edge states of quantum-Hall-effect (QHE) systems, and are also immune to backscattering and localization by disorder. The Berry curvature of the photonic bands plays a role analogous to that of the magnetic field in the QHE. Explicit calculations demonstrating the existence of such unidirectionally propagating photonic edge states are presented.

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