Communications on Pure and Applied Mathematics最新文献

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The α$alpha$‐SQG patch problem is illposed in C2,β$C^{2,beta }$ and W2,p$W^{2,p}$ 在 C2,β$C^{2,beta }$ 和 W2,p$W^{2,p}$ 中，α$alpha$-SQG 补丁问题存在问题。

IF 3 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1002/cpa.22236

Alexander Kiselev, Xiaoyutao Luo

We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in every Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for every Sobolev space unless .

我们考虑的是-（表面准地养）SQG 系统的补集问题，其值和分别为二维欧拉方程和 SQG 方程。众所周知，欧拉补集在非端点荷尔德空间以及在Ⅳ空间中都是全局良好的。此外，在合适的正则范围内，除非......，否则每个 Sobolev 空间都具有相同的强失稳性。

引用次数: 0

Mean‐field limit of non‐exchangeable systems 不可交换系统的均场极限

IF 3 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1002/cpa.22235

Pierre‐Emmanuel Jabin, David Poyato, Juan Soler

This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.

本文论述了一大类稀疏图上多代理系统均场极限的推导。更具体地说，本文探讨了由非相同代理组成的不可交换多代理系统的情况。分析不仅涉及 PDEs 和随机分析，还通过稀疏图（扩展图子）极限的新概念涉及图论，这反映了网络中的连接性结构，并对集体动力学产生了关键影响。在这篇文章中，以往文献中关于代理之间的连通性（密集图）和代理之间的合作（对称互动）的一些主要限制性假设被删除了。

引用次数: 0

Semiconvexity estimates for nonlinear integro‐differential equations 非线性积分微分方程的半凸性估计

IF 3 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-11-15 DOI: 10.1002/cpa.22237

Xavier Ros‐Oton, Clara Torres‐Latorre, Marvin Weidner

In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.

在本文中，我们首次建立了全非线性方程和由具有一般核的整微分算子驱动的障碍问题的局部半凸性估计。我们的证明基于伯恩斯坦技术，该技术是为一类自然的非局部算子开发的，并被认为具有独立的意义。特别是，我们解决了卡布雷-迪皮耶罗-瓦尔迪诺奇的一个未决问题。作为我们结果的应用，我们为域上的非局部障碍问题建立了最优正则性估计和正则点附近自由边界的平滑性。最后，我们还将伯恩斯坦技术扩展到抛物方程和非对称算子。

引用次数: 0

Convergence to the planar interface for a nonlocal free-boundary evolution 收敛到非局部自由边界演化的平面界面

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-09-05 DOI: 10.1002/cpa.22225

Felix Otto, Richard Schubert, Maria G. Westdickenberg

We capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well-prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one-dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

我们捕捉 Mullins-Sekerka 演化的最佳衰减，这是材料科学中的一个非局部抛物自由边界问题。我们的主要结果证明，在环境空间维数为三的物理相关情况下，BV 解收敛于平面轮廓。我们不假定初始数据较小或准备充分，而是允许初始界面不具有图形结构且不相连，因此明确包括奥斯特瓦尔德熟化机制。仅就初始有限（不小于）过剩质量和过剩表面能而言，我们确定表面在一个固定的时间尺度（定量估计）内成为一个 Lipschitz 图形，并在此环境中保持困顿。为了获得图结构，我们利用了几何度量理论的正则性结果。同时，我们将以前用于一维 PDE 问题的对偶方法扩展到了更高维度的非局部几何演化。我们获得了过剩能量、耗散和图高度的最佳代数衰减率。

引用次数: 0

Asymptotics of block Toeplitz determinants with piecewise continuous symbols 具有片断连续符号的块托普利兹行列式的渐近论

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-08-28 DOI: 10.1002/cpa.22223

Estelle Basor, Torsten Ehrhardt, Jani A. Virtanen

We determine the asymptotics of the block Toeplitz determinants $� � � det � �_{T � n �} � � (� ϕ �) � � � �$ as $� � � n � \to � \infty � � �$ for $� � � N � \times � N � � �$ matrix-valued piecewise continuous functions $� � ϕ � �$ with a finitely many jumps under mild additional conditions. In particular, we prove that

在温和的附加条件下，我们确定了块托普利兹行列式的渐近线，如同具有有限次跳跃的矩阵值片断连续函数。特别是，我们证明了，，和是取决于矩阵符号的常数，并在我们的主要结果中进行了描述。我们的方法基于托普利兹行列式的新局部定理、计算具有片断连续矩阵值符号的托普利兹算子的弗雷德霍姆指数的新方法以及其他算子理论方法。作为我们结果的一个应用，我们考虑了在量子自旋链模型的纠缠熵研究中出现的片断连续符号。

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引用次数: 0

Tight Lipschitz hardness for optimizing mean field spin glasses 优化均值场自旋玻璃的严格 Lipschitz 硬度

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-08-20 DOI: 10.1002/cpa.22222

Brice Huang, Mark Sellke

We study the problem of algorithmically optimizing the Hamiltonian <math> <semantics> <msub> <mi>H</mi> <mi>N</mi> </msub> <annotation>$H_N$</annotation> </semantics></math> of a spherical or Ising mixed <math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math>-spin glass. The maximum asymptotic value <math> <semantics> <mi>OPT</mi> <annotation>${mathsf {OPT}}$</annotation> </semantics></math> of <math> <semantics> <mrow> <msub> <mi>H</mi> <mi>N</mi> </msub> <mo>/</mo> <mi>N</mi> </mrow> <annotation>$H_N/N$</annotation> </semantics></math> is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing (AMP) algorithms efficiently optimize <math> <semantics> <mrow> <msub> <mi>H</mi> <mi>N</mi> </msub> <mo>/</mo> <mi>N</mi> </mrow> <annotation>$H_N/N$</annotation> </semantics></math> up to a value <math> <semantics> <mi>ALG</mi> <annotation>${mathsf {ALG}}$</annotation> </semantics></math> given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property (OGP). However, <math> <semantics> <mrow> <mi>ALG</mi> <mo><</mo> <mi>OPT</mi> </mrow> <annotation>${mathsf {ALG}}< {mathsf {OPT}}$</annotation> </semantics></math> can also occur, and no efficient algorithm producing an objective value exceeding <math> <semantics> <mi>ALG</mi> <annotation>${mathsf {ALG}}$</annotation> </semantics></math> is known. We prove that for mixed even <math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math>-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than <math> <semantics> <mi>ALG</mi> <annotation>${mathsf {ALG}}$</annotation> </semantics></math> with non-negligible probability.

我们研究了球面或伊辛混合 p $p$ -自旋玻璃的哈密顿H N $H_N$ 的算法优化问题。H N / N $H_N/N$ 的最大渐近值 OPT ${mathsf {OPT}}$ 是由一个称为帕里西公式的变分原理表征的。最近开发的近似消息传递（AMP）算法可以有效优化 H N / N $H_N/N$ 达到扩展帕里西公式给出的值 ALG ${mathsf {ALG}}$，该值在更大的功能阶参数空间上最小化。对于表现出无重叠间隙特性（OGP）的自旋玻璃来说，这两个目标是相等的。然而，ALG < OPT ${mathsf {ALG}}< {mathsf {OPT}}$ 也可能出现，而且目前还不知道哪种高效算法能产生超过 ALG ${mathsf {ALG}}$ 的目标值。我们证明，对于混合偶数 p $p$ -自旋模型，没有一种满足重叠集中特性的算法能以不可忽略的概率产生大于 ALG ${mathsf {ALG}}$的目标值。这一特性适用于所有对 H N $H_N$ 的无序系数具有适当 Lipschitz 依赖性的算法。它包括梯度下降、AMP 和朗格文动力学在有界时间内运行的自然公式，尤其包括上述实现 ALG ${mathsf {ALG}}$的算法。为了证明这一结果，我们将 Gamarnik 和 Sudan 引入的 OGP 框架大幅推广到任意超对称禁止解结构。

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引用次数: 0

Global regularity for critical SQG in bounded domains 有界域中临界 SQG 的全局正则性

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-07-24 DOI: 10.1002/cpa.22221

Peter Constantin, Mihaela Ignatova, Quoc-Hung Nguyen

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in $� � �^{R � 2 �} � �$ . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

我们证明了有界域中临界耗散 SQG 方程全局平稳解的存在性和唯一性。我们引入了一种新方法，将有界域中的单一非局部非线性演化方程转化为整个空间中的扩展非局部非线性演化方程的相互作用系统。然后利用扩展系统中的非局部算子的非线性最大原理方法进行证明。

引用次数: 0

A variational construction of Hamiltonian stationary surfaces with isolated Schoen–Wolfson conical singularities 具有孤立 Schoen-Wolfson 圆锥奇点的哈密顿静止面的变分构造

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-06-07 DOI: 10.1002/cpa.22220

Filippo Gaia, Gerard Orriols, Tristan Rivière

We construct using variational methods Hamiltonian stationary surfaces with isolated Schoen–Wolfson conical singularities. We obtain these surfaces through a convergence process reminiscent to the Ginzburg–Landau asymptotic analysis in the strongly repulsive regime introduced by Bethuel, Brezis and Hélein. We describe in particular how the prescription of Schoen–Wolfson conical singularities is related to optimal Wente constants.

我们利用变分法构建了具有孤立肖恩-沃尔夫森圆锥奇点的哈密顿静止曲面。我们通过一个收敛过程来获得这些表面，这个过程让人联想到 Bethuel、Brezis 和 Hélein 提出的强排斥机制中的金兹堡-兰道渐近分析。我们特别描述了舍恩-沃尔夫森锥奇点的处方与最佳温特常数之间的关系。

引用次数: 0

Almost sharp lower bound for the nodal volume of harmonic functions 谐函数节点体积的近似尖锐下界

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-05-29 DOI: 10.1002/cpa.22207

Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $� � u � �$ be a real-valued harmonic function in $� � �^{R � n �} � �$ with $� � � u � (� 0 �) � = � 0 � � �$ and $� � � n � \geq � 3 � � �$ . We prove

本文主要研究谐函数的增长与其零集的 Hausdorff 度量之间的关系。设是一个实值谐函数，且。我们证明了翻倍指数是由定义的增长概念，这给出了、的零集的 Hausdorff 度量的一个近乎尖锐的下限，猜想它是线性的。文章的新内容是稳定增长的概念，以及谐函数倍指数分布下界的多尺度归纳技术。与之前最著名的下界，即纳迪拉什维利猜想相比，它给出了一个重大改进。

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引用次数: 0

Allen–Cahn solutions with triple junction structure at infinity 无穷远处具有三重结点结构的艾伦-卡恩解

IF 3.1 1区数学 Q1 MATHEMATICS

Communications on Pure and Applied Mathematics

Pub Date : 2024-05-17 DOI: 10.1002/cpa.22204

Étienne Sandier, Peter Sternberg

We construct an entire solution $� � � U � : � �^{R � 2 �} � \to � �^{R � 2 �} � � �$ to the elliptic system

我们构建了一个椭圆系统的整体解，其中有一个 "三井 "势。这个解是相关能量的局部最小化，即在任何紧凑集合上，与该集合外的竞争者一致的能量最小化。此外，我们还证明，沿着子序列，"三井 "的 "井喷 "会逼近一个最小的 "三井"，即......。以前的结果假设了不同程度的势对称性，并没有建立局部最小性，但在这里我们不做这样的对称性假设。

引用次数: 0

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