Communications on Pure and Applied Mathematics最新文献_第10页

Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the ( n − 2 ) $(n-2)$ -area functional 自对偶U(1)‐Yang-Mills-Higgs能量收敛到(n−2)$(n-2)$‐面积泛函

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-07 DOI: 10.1002/cpa.22150

Davide Parise, Alessandro Pigati, Daniel Stern

引用次数: 0

C 2 , α $C^{2,alpha }$ regularity of free boundaries in optimal transportation 最优运输中自由边界的C2，α$C^{2，alpha}$正则性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-07 DOI: 10.1002/cpa.22151

Shibing Chen, Jiakun Liu, Xu-Jia Wang

引用次数: 0

Prescribed curvature measure problem in hyperbolic space 双曲空间中的曲率测度问题

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-07 DOI: 10.1002/cpa.22160

Fengrui Yang

引用次数: 4

Free boundary partial regularity in the thin obstacle problem 薄障碍问题的自由边界部分正则性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-05 DOI: 10.1002/cpa.22152

Federico Franceschini, Joaquim Serra

{"title":"Free boundary partial regularity in the thin obstacle problem","authors":"Federico Franceschini, Joaquim Serra","doi":"10.1002/cpa.22152","DOIUrl":"10.1002/cpa.22152","url":null,"abstract":"For the thin obstacle problem in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^n$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$nge 2$</annotation>\u0000 </semantics></math>, we prove that at all free boundary points, with the exception of a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>3</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(n-3)$</annotation>\u0000 </semantics></math>-dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a C1, 1-type free boundary regularity result, up to a codimension 3 set.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"630-669"},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43420067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Existence of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics 理想可压缩磁流体力学的多维接触间断性的存在性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-05 DOI: 10.1002/cpa.22148

Yanjin Wang, Zhouping Xin

{"title":"Existence of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics","authors":"Yanjin Wang, Zhouping Xin","doi":"10.1002/cpa.22148","DOIUrl":"10.1002/cpa.22148","url":null,"abstract":"We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well-posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"583-629"},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43933569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Hearing the shape of ancient noncollapsed flows in R 4 $mathbb {R}^{4}$ 在R4$mathbb {R}^{4}$中听到古代非崩塌流的形状

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-05 DOI: 10.1002/cpa.22140

Wenkui Du, Robert Haslhofer

{"title":"Hearing the shape of ancient noncollapsed flows in \u0000 \u0000 \u0000 R\u0000 4\u0000 \u0000 $mathbb {R}^{4}$","authors":"Wenkui Du, Robert Haslhofer","doi":"10.1002/cpa.22140","DOIUrl":"10.1002/cpa.22140","url":null,"abstract":"We consider ancient noncollapsed mean curvature flows in R4$mathbb {R}^4$ whose tangent flow at −∞$-infty$ is a bubble‐sheet. We carry out a fine spectral analysis for the bubble‐sheet function u that measures the deviation of the renormalized flow from the round cylinder R2×S1(2)$mathbb {R}^2 times S^1(sqrt {2})$ and prove that for τ→−∞$tau rightarrow -infty$ we have the fine asymptotics u(y,θ,τ)=(y⊤Qy−2tr(Q))/|τ|+o(|τ|−1)$u(y,theta ,tau )= (y^top Qy -2textrm {tr}(Q))/|tau | + o(|tau |^{-1})$ , where Q=Q(τ)$Q=Q(tau )$ is a symmetric 2 × 2‐matrix whose eigenvalues are quantized to be either 0 or −1/8$-1/sqrt {8}$ . This naturally breaks up the classification problem for general ancient noncollapsed flows in R4$mathbb {R}^4$ into three cases depending on the rank of Q. In the case rk(Q)=0$mathrm{rk}(Q)=0$ , generalizing a prior result of Choi, Hershkovits and the second author, we prove that the flow is either a round shrinking cylinder or R×$mathbb {R}times$ 2d‐bowl. In the case rk(Q)=1$mathrm{rk}(Q)=1$ , under the additional assumption that the flow either splits off a line or is self‐similarly translating, as a consequence of recent work by Angenent, Brendle, Choi, Daskalopoulos, Hershkovits, Sesum and the second author we show that the flow must be R×$mathbb {R}times$ 2d‐oval or belongs to the one‐parameter family of 3d oval‐bowls constructed by Hoffman‐Ilmanen‐Martin‐White, respectively. Finally, in the case rk(Q)=2$mathrm{rk}(Q)=2$ we show that the flow is compact and SO(2)‐symmetric and for τ→−∞$tau rightarrow -infty$ has the same sharp asymptotics as the O(2) × O(2)‐symmetric ancient ovals constructed by Hershkovits and the second author. The full classification problem will be addressed in subsequent papers based on the results of the present paper.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"543-582"},"PeriodicalIF":3.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45668603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

On the Maxwell-Bloch system in the sharp-line limit without solitons 无孤子的直线极限下麦克斯韦-布洛赫系统

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-08-21 DOI: 10.1002/cpa.22136

Sitai Li, Peter D. Miller

{"title":"On the Maxwell-Bloch system in the sharp-line limit without solitons","authors":"Sitai Li, Peter D. Miller","doi":"10.1002/cpa.22136","DOIUrl":"10.1002/cpa.22136","url":null,"abstract":"We study the (characteristic) Cauchy problem for the Maxwell-Bloch equations of light-matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically-motivated initial-boundary conditions are satisfied. In particular, we present a proper Riemann-Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading-order asymptotics to self-similar solutions that satisfy a system of ordinary differential equations related to the Painlevé-III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self-similar solutions. We make this observation precise and for the first time we relate the PIII self-similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially-unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"457-542"},"PeriodicalIF":3.0,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44206925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

A generalization of Geroch's conjecture Geroch猜想的一个推广

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-08-12 DOI: 10.1002/cpa.22137

Simon Brendle, Sven Hirsch, Florian Johne

{"title":"A generalization of Geroch's conjecture","authors":"Simon Brendle, Sven Hirsch, Florian Johne","doi":"10.1002/cpa.22137","DOIUrl":"10.1002/cpa.22137","url":null,"abstract":"The Theorem of Bonnet–Myers implies that manifolds with topology <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>M</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$M^{n-1} times mathbb {S}^1$</annotation>\u0000 </semantics></math> do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {T}^n$</annotation>\u0000 </semantics></math> does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so-called m-intermediate curvature), and use stable weighted slicings to show that for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≤</mo>\u0000 <mn>7</mn>\u0000 </mrow>\u0000 <annotation>$n le 7$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>m</mi>\u0000 <mo>≤</mo>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$1 le m le n-1$</annotation>\u0000 </semantics></math> the manifolds <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>M</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$N^n = M^{n-m} times mathbb {T}^m$</annotation>\u0000 </semantics></math> do not admit a metric of positive m-intermediate curvature.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"441-456"},"PeriodicalIF":3.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49280234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 7

Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension 有限渐近维均匀可收缩流形上标量曲率的衰减

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-08-02 DOI: 10.1002/cpa.22128

Jinmin Wang, Zhizhang Xie, Guoliang Yu

引用次数: 11

Stability of the tangent bundle through conifold transitions 通过针叶树跃迁的切丛的稳定性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-07-28 DOI: 10.1002/cpa.22135

Tristan Collins, Sebastien Picard, Shing-Tung Yau

{"title":"Stability of the tangent bundle through conifold transitions","authors":"Tristan Collins, Sebastien Picard, Shing-Tung Yau","doi":"10.1002/cpa.22135","DOIUrl":"10.1002/cpa.22135","url":null,"abstract":"Let X be a compact, Kähler, Calabi-Yau threefold and suppose <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>↦</mo>\u0000 <munder>\u0000 <mi>X</mi>\u0000 <mo>̲</mo>\u0000 </munder>\u0000 <mo>⇝</mo>\u0000 <msub>\u0000 <mi>X</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$Xmapsto underline{X}leadsto X_t$</annotation>\u0000 </semantics></math> , for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>∈</mo>\u0000 <mi>Δ</mi>\u0000 </mrow>\u0000 <annotation>$tin Delta$</annotation>\u0000 </semantics></math>, is a conifold transition obtained by contracting finitely many disjoint <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(-1,-1)$</annotation>\u0000 </semantics></math> curves in X and then smoothing the resulting ordinary double point singularities. We show that, for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>t</mi>\u0000 <mo>|</mo>\u0000 <mo>≪</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$|t|ll 1$</annotation>\u0000 </semantics></math> sufficiently small, the tangent bundle <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msup>\u0000 <msub>\u0000 <mi>X</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$T^{1,0}X_{t}$</annotation>\u0000 </semantics></math> admits a Hermitian-Yang-Mills metric <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <annotation>$H_t$</annotation>\u0000 </semantics></math> with respect to the conformally balanced metrics constructed by Fu-Li-Yau. Furthermore, we describe the behavior of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <annotation>$H_t$</annotation>\u0000 </semantics></math> near the vanishing cycles of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>X</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <annotation>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 1","pages":"284-371"},"PeriodicalIF":3.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41357784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 14