Journal de l’École polytechnique — Mathématiques最新文献

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Large mass rigidity for a liquid drop model in 2D with kernels of finite moments 具有有限矩核的二维液滴模型的大质量刚度

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-06-04 DOI: 10.5802/jep.178

B. Merlet, Marc Pegon

. — Motivated by Gamow’s liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter P ( E ) is replaced by P ( E ) − γP ε ( E ) , with 0 < γ < 1 and P ε a nonlocal energy such that P ε ( E ) → P ( E ) as ε vanishes. We prove that unit area minimizers are disks for ε > 0 small enough. More precisely, we ﬁrst show that in dimension 2 , minimizers are necessarily convex, provided that ε is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension n (cid:62) 2 ) the unit ball in R n is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, suﬃciently integrable kernel. In that formulation, our main result states that if the ﬁrst moment of the kernel is smaller than an explicit threshold, there exists a critical mass m 0 such that for any m > m 0 , the disk is the unique minimizer of area m up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass. ) − γP ε ( E ) , où 0 < γ < 1 et P ε est une énergie non locale telle que P ε ( E ) → P ( E ) lorsque ε tend vers zéro. Nous montrons que pour ε assez petit les minimiseurs à aire ﬁxée sont les disques. Pour cela, nous établissons d’abord qu’en dimension 2 , les minimiseurs sont convexes dès que ε est suﬃsamment petit. Ceci implique que le bord d’un minimiseur est une petite perturbation Lipschitz d’un cercle. Puis, par un argument à la Fuglede, nous prouvons (en dimension arbitraire n (cid:62) 2 ) que si un minimiseur à volume ﬁxé est une perturbation d’une boule au sens précédent, alors c’est une boule. Ce problème isopérimétrique est équivalent à une généralisation du modèle de goutte liquide pour le noyau atomique introduit par Gamow lorsque le potentiel répulsif non local est donné par un noyau suﬃsamment intégrable. Dans cette formulation, notre résultat principal indique que si le premier moment du noyau est inférieur à un seuil explicite, il existe une masse critique m 0 telle que les minimiseurs de masse prescrite m > m 0 sont les disques. Ceci contraste fortement avec le cas classique des noyaux de Riesz, où le problème n’admet pas de minimiseur au-delà d’une masse critique.

．基于大质量区域的伽莫夫液滴模型，我们考虑了一个等周问题，其中标准周长P (E)被P (E)−γP ε (E)所取代，且P ε为非局域能量，使得P ε (E)→P (E)作为ε消失。我们证明了ε > 0足够小的单位面积最小值是圆盘。更准确地说，我们首先证明了在维数2中，只要ε足够小，最小值必然是凸的。反过来，这意味着极小值具有近似圆形的边界，也就是说，它们的边界是圆的一个小的利普希茨摄动。然后，利用fuglede型论证，证明了(在任意维数n (cid:62) 2) R n中的单位球是该问题在中心近球集中唯一的单位体积最小值。因此，在翻译之前，单元磁盘是唯一的最小化器。这个等周问题等价于伽莫夫引入的原子核液滴模型的推广，其中非局部排斥势由一个径向的、充分可积的核给出。在该公式中，我们的主要结果表明，如果核的第一个矩小于一个显式阈值，则存在一个临界质量m 0，使得对于任何m > m 0，磁盘是面积m的唯一最小值，直到平移。这与Riesz核的通常情况形成鲜明对比，在Riesz核中，问题不允许超过临界质量的最小值。)−γP ε (E)， où 0 < γ < 1 et P ε est une samengie non locale telle que P ε (E)→P (E) lorsque ε tend vers zsamenro。我不知道你是谁，我不知道你是谁，我不知道你是谁。倒cela，在维数为2的情况下，最小的情况下，凸的情况下，最小的情况下，最小的情况下，最小的情况下，最小的情况下，最小的情况下，最小的情况是最小的。李普希兹环的最小摄动。如:(1)根据Fuglede的论点，任意维数为1 (cid:62) 2)的任意维数为1(最小体积固定)的任意维数为1，任意维数为1，任意维数为1，任意维数为1，任意维数为1，任意维数为1。问题是:isopsamrim - samtrique - est - samequivalent - est - samequivalent - est - est - est - est - est - est - est - est - est - est - est - est如果有一种新的表述，即“没有任何一种主要的个体形式”，即“没有任何一种主要的个体形式”，即“没有任何一种主要的个体形式”，那么“没有任何一种主要的个体形式”，即“没有任何一种主要的个体形式”，即“没有任何一种主要的个体形式”。Ceci对比了两种观点，一种是经典的观点，一种是问题的观点，另一种是大众批评的观点。

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引用次数: 6

Coexistence of chaotic and elliptic behaviors among analytic, symplectic diffeomorphisms of any surface 任意曲面的解析、交点衍射之间的混沌和椭圆行为共存

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-05-18 DOI: 10.5802/jep.224

P. Berger

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982).

我们证明了在任何封闭曲面的解析交映衍射中，混沌行为（正度量熵）和椭圆行为（可内卷的 KAM 岛）是共存的。特别是，这解决了 F. Przytycki（1982 年）提出的一个问题。

引用次数: 3

Zero-cycle groups on algebraic varieties 代数变异上的零环群

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-04-16 DOI: 10.5802/jep.183

F. Binda, A. Krishna

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of 0-cycles over p-adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.

我们比较了域上拟射影变异上的各种0环群。作为应用，我们证明了对于某些奇异射影变，0环的Levine-Weibel Chow群与相应的Friedlander-Voevodsky动机上同。我们还证明了在一个具有正特征的代数闭域上，关于一个约除数的光滑射影变化上具有模的0环的Chow群与该除数的补的Suslin同调是一致的。证明了p进域上0环的Chow群的Saito和Sato有限定理的几个推广。我们还利用这些结果推导出了Suslin同调的一个扭转定理，该定理将Bloch的结果推广到开变。

引用次数: 7

Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation 从下面的密度估计与a . Zygmund关于Lipschitz微分的猜想有关

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-04-10 DOI: 10.5802/jep.211

T. Pauw

Letting $A subset mathbb{R}^n$ be Borel measurable and $W_0 : A to mathbb{G}(n,m)$ Lipschitzian, we establish that begin{equation*} limsup_{r to 0^+} frac{mathcal{H}^m left[ A cap B(x,r) cap (x+ W_0(x))right]}{alpha(m)r^m} geq frac{1}{2^n}, end{equation*} for $mathcal{L}^n$-almost every $x in A$. In particular, it follows that $A$ is $mathcal{L}^n$-negligible if and only if $mathcal{H}^m(A cap (x+W_0(x))=0$, for $mathcal{L}^n$-almost every $x in A$.

假设$A subset mathbb{R}^n$是Borel可测量的，并且$W_0 : A to mathbb{G}(n,m)$是Lipschitzian的，我们建立了begin{equation*} limsup_{r to 0^+} frac{mathcal{H}^m left[ A cap B(x,r) cap (x+ W_0(x))right]}{alpha(m)r^m} geq frac{1}{2^n}, end{equation*}对于$mathcal{L}^n$——几乎所有$x in A$。特别地，可以得出结论:$A$等于$mathcal{L}^n$——当且仅当$mathcal{H}^m(A cap (x+W_0(x))=0$对于$mathcal{L}^n$几乎等于$x in A$时可以忽略不计。

引用次数: 0

Quantitative De Giorgi methods in kinetic theory 动态理论中的定量德乔基方法

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-03-17 DOI: 10.5802/jep.203

Jessica Guerand, C. Mouhot

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{"o}lder continuity with quantitative estimates. The paper is self-contained.

我们考虑在漂移-扩散算子中具有粗糙系数的动力学Fokker-Planck型准椭圆方程，也称为Kolmogorov或超抛物线方程。给出了De Giorgi中值引理以及弱哈纳克不等式和哈纳克不等式的简短定量证明。这意味着H{ ' o}与定量估计的更大连续性。这张纸是独立的。

引用次数: 23

Quantitative homogenization theory for random suspensions in steady Stokes flow 稳定Stokes流中随机悬浮的定量均匀化理论

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-03-11 DOI: 10.5802/jep.204

Mitia Duerinckx, A. Gloria

Abstract. This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.

摘要本工作发展了稳定斯托克斯流中刚性颗粒随机悬浮的定量均匀化理论，并完成了最近的定性结果。更准确地说，我们建立了Stokes问题的大尺度正则性理论，并证明了相关校正量的矩界和均匀化误差的最优估计;后者进一步要求对随机悬架进行定量遍历假设。与发散型线性椭圆方程的定量均匀化理论相比，对流体不可压缩性和粒子刚性约束的分析存在很大的困难。我们的分析进一步适用于(可压缩或不可压缩)线弹性和静电中的刚性内含物问题;在这些情况下，即使在周期性设置中，它也是新的。

引用次数: 11

Product set growth in Burnside groups 伯恩赛德集团的产品集增长

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-02-22 DOI: 10.5802/jep.187

Rémi Coulon, M. Steenbock

A BSTRACT . Given a periodic quotient of a torsion-free hyperbolic group, we provide a ﬁne lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Saﬁn for free groups. 20F65, 20F67, 20F50, 20F06, 20F69.

摘要。给出了一个无扭双曲群的周期商，给出了任意子半群的生长函数的一个精细下估计。这推广了Razborov和Safin对自由群体的结果。20f65、20f67、20f50、20f06、20f69。

引用次数: 3

Birkhoff normal forms for Hamiltonian PDEs in their energy space 哈密顿偏微分方程在能量空间中的Birkhoff范式

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-02-19 DOI: 10.5802/jep.193

J. Bernier, B. Gr'ebert

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a strong enough energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that, to exchange energy, modes have to oscillate at the same frequency. Contrary to the previous existing results, we do not require the solutions to be especially smooth. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schr{"o}dinger equations in dimension d $le$ 2.

研究了半线性色散哈密顿偏微分方程小解在受限域上的长时性。假设系统具有新的非共振条件和足够强的能量估计，我们证明了它的低超作用几乎保持了很长时间。粗略地说，这意味着要交换能量，模态必须以相同的频率振荡。与之前已有的结果相反，我们并不要求解特别光滑。它们只需要生活在能量空间。我们将所得结果应用于维数d = 1的非线性Klein-Gordon方程和维数d $ le$ 2的非线性Schr{ ' o}dinger方程。

引用次数: 5

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces 球锥表面间极小拉格朗日微分同态的刚性

Journal de l’École polytechnique — Mathématiques

Pub Date : 2021-01-18 DOI: 10.5802/jep.190

Christian El Emam, Andrea Seppi

—We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions. Résumé (Rigidité des difféomorphismes minimaux lagrangiens entre surfaces sphériques à singularités coniques) Nous démontrons que toute application minimale lagrangienne entre deux surfaces fermées sphériques à singularités coniques est une isométrie, sans aucune hypothèse sur les valeurs des multi-angles des deux surfaces. En appliquant ce résultat, nous prouvons une généralisation du théorème classique de rigidité de Liebmann, notamment l’énoncé que toute immersion dans l’espace euclidien de dimension 3 d’une surface fermée avec courbure gaussienne constante positive et avec points de ramification est un revêtement ramifié sur une sphère.

-我们证明了两个具有圆锥体奇点的封闭球面之间的任何最小拉格朗日差异是等距的，没有任何关于两个曲面多角度的假设。we show that, As an应用叫支学院关闭了表面浸入恒正Gaussian curvature in Euclidean three-space is a支,到a轮,thus generalizing the sphere”rigidity theorem of Liebmann to支倾倒。摘要(圆锥奇点球面之间的拉格朗日最小差异的刚度)我们证明了两个圆锥奇点球面之间的任何拉格朗日最小映射都是等距的，对两个曲面的多角度值没有任何假设。应用这一结果，我们证明了经典的Liebmann刚度定理的推广，特别是具有正高斯常数曲率和分支点的封闭曲面浸入欧几里得3维空间是球上的分支覆盖。

引用次数: 0

Unconditional Chebyshev biases in number fields 数域中的无条件切比雪夫偏置

Journal de l’École polytechnique — Mathématiques

Pub Date : 2020-12-22 DOI: 10.5802/jep.192

D. Fiorilli, F. Jouve

. Prime counting functions are believed to exhibit, in various contexts, discrep-ancies beyond what famous equidistribution results predict; this phenomenon is known as Chebyshev’s bias. Rubinstein and Sarnak have developed a framework which allows to con-ditionally quantify biases in the distribution of primes in general arithmetic progressions. Their analysis has been generalized by Ng to the context of the Chebotarev density theorem, under the assumption of the Artin holomorphy conjecture, the Generalized Riemann Hypothesis, as well as a linear independence hypothesis on the zeros of Artin L -functions. In this paper we show unconditionally the occurence of extreme biases in this context. These biases lie far beyond what the strongest eﬀective forms of the Chebotarev density theorem can predict. More precisely, we prove the existence of an inﬁnite family of Galois extensions and associated conjugacy classes C 1 , C 2 ⊂ Gal( L/K ) of same size such that the number of prime ideals of norm up to x with Frobenius conjugacy class C 1 always exceeds that of Frobenius conjugacy class C 2 , for every large enough x . A key argument in our proof relies on features of certain subgroups of symmetric groups which enable us to circumvent the need for unproven properties of zeros of Artin L -functions.

．素数计数函数被认为在各种情况下表现出超出著名的均匀分布结果所预测的差异;这种现象被称为切比雪夫偏差。Rubinstein和Sarnak开发了一个框架，该框架允许有条件地量化一般等差数列中素数分布中的偏差。他们的分析被Ng推广到Chebotarev密度定理的背景下，在Artin全纯猜想的假设下，在广义黎曼假设下，以及在Artin L -函数的零点上的线性无关假设。在本文中，我们无条件地证明了在这种情况下极端偏差的发生。这些偏差远远超出了切波塔列夫密度定理最有效的形式所能预测的范围。更确切地说，我们证明了同样大小的无限一族伽罗瓦扩展及其共轭类c1, c2∧Gal(L/K)的存在性，使得对于每一个足够大的x，具有Frobenius共轭类c1的最大范数的素数理想的个数总是超过Frobenius共轭类c2的素数理想的个数。我们证明中的一个关键论点依赖于对称群的某些子群的特征，这些特征使我们能够规避对Artin L -函数的零的未证明性质的需要。

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引用次数: 5

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Journal de l’École polytechnique — Mathématiques

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