Communications on Pure and Applied Mathematics最新文献

{"title":"Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature","authors":"Gioacchino Antonelli, Marco Pozzetta, Daniele Semola","doi":"10.1002/cpa.22252","DOIUrl":"https://doi.org/10.1002/cpa.22252","url":null,"abstract":"Let be a complete Riemannian manifold which is not isometric to , has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set with density 1 at infinity such that for every there is a unique isoperimetric set of volume in ; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals such that isoperimetric sets with volumes exist, but they are neither unique nor do they have strictly volume preserving stable boundaries.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"73 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143782633","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}

{"title":"The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion","authors":"Benjamin Gess, Daniel Heydecker","doi":"10.1002/cpa.22251","DOIUrl":"https://doi.org/10.1002/cpa.22251","url":null,"abstract":"The problem of deriving a gradient flow structure for the porous medium equation which is <jats:italic>thermodynamic</jats:italic>, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero‐range process with jump rate is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The key super‐exponential estimate is obtained using pathwise discretised regularity estimates in the spirit of the Aubin–Lions–Simons lemma. This allows to exhibit the porous medium equation as the gradient flow of the entropy in a thermodynamic metric via the energy‐dissipation inequality.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"34 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143782632","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}

{"title":"First‐order Sobolev spaces, self‐similar energies and energy measures on the Sierpiński carpet","authors":"Mathav Murugan, Ryosuke Shimizu","doi":"10.1002/cpa.22247","DOIUrl":"https://doi.org/10.1002/cpa.22247","url":null,"abstract":"For any , we construct ‐energies and the corresponding ‐energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self‐similarity of energy. An important motivation for the construction of self‐similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal dimension is attained, we show that any optimal Ahlfors regular measure attaining the Ahlfors regular conformal dimension must necessarily be a bounded perturbation of the ‐energy measure of some function in our Sobolev space, where is the Ahlfors regular conformal dimension. Under the attainment of the Ahlfors regular conformal dimension, the ‐Newtonian Sobolev space corresponding to any optimal Ahlfors regular metric and measure is shown to coincide with our Sobolev space with comparable norms, where is the Ahlfors regular conformal dimension.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"12 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143443282","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}

{"title":"On the Read-Shockley energy for grain boundaries in 2D polycrystals","authors":"Martino Fortuna,&nbsp;Adriana Garroni,&nbsp;Emanuele Spadaro","doi":"10.1002/cpa.22245","DOIUrl":"10.1002/cpa.22245","url":null,"abstract":"<p>In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearized elasticity and an ansatz on the distribution of incompatibilities of the lattice at the interface between two grains. The logarithmic scaling of this formula has been rigorously justified without any ansatz on the geometry of dislocations only recently in an article by Lauteri and Luckhaus. In the present paper, building upon their analysis, we derive a two dimensional sharp interface limiting functional starting from the nonlinear semi-discrete model introduced in Lauteri and Luckhaus: the line tension we obtain via <span></span><math>\u0000 <semantics>\u0000 <mi>Γ</mi>\u0000 <annotation>$Gamma$</annotation>\u0000 </semantics></math>-convergence depends on the rotations of the grains and the relative orientations of the interfaces, and for small angle grain boundaries has the Read and Shockley logarithmic scaling.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 8","pages":"1411-1459"},"PeriodicalIF":3.1,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143384980","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}

{"title":"Stability of perfectly matched layers for Maxwell's equations in rectangular solids","authors":"Laurence Halpern,&nbsp;Jeffrey Rauch","doi":"10.1002/cpa.22249","DOIUrl":"10.1002/cpa.22249","url":null,"abstract":"<p>Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$mathbb {R}^{1+3}$</annotation>\u0000 </semantics></math> using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annotation>$tau$</annotation>\u0000 </semantics></math>, the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When <span></span><math>\u0000 <semantics>\u0000 <mi>τ</mi>\u0000 <annotation>$tau$</annotation>\u0000 </semantics></math> is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$H^2$</annotation>\u0000 </semantics></math> solutions with uniform <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^2$</annotation>\u0000 </semantics></math> estimates. Using the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$H^2$</annotation>\u0000 </semantics></math> regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison-Kenig-Mitrea is extended to compensate the nonpositivity of the flux.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 8","pages":"1460-1518"},"PeriodicalIF":3.1,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143393170","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}

{"title":"Analysis of density matrix embedding theory around the non-interacting limit","authors":"Eric Cancès,&nbsp;Fabian M. Faulstich,&nbsp;Alfred Kirsch,&nbsp;Eloïse Letournel,&nbsp;Antoine Levitt","doi":"10.1002/cpa.22244","DOIUrl":"10.1002/cpa.22244","url":null,"abstract":"<p>This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-representability problem inherent to DMET.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 8","pages":"1359-1410"},"PeriodicalIF":3.1,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22244","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143125219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Special Lagrangian pair of pants","authors":"Yang Li","doi":"10.1002/cpa.22248","DOIUrl":"10.1002/cpa.22248","url":null,"abstract":"<p>We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>T</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$T^n$</annotation>\u0000 </semantics></math> with the Euclidean structure, building upon earlier topological ideas of Matessi. The construction uses a combination of PDE and geometric measure theory.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 7","pages":"1320-1356"},"PeriodicalIF":3.1,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143056314","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}

{"title":"Localized and degenerate controls for the incompressible Navier–Stokes system","authors":"Vahagn Nersesyan,&nbsp;Manuel Rissel","doi":"10.1002/cpa.22246","DOIUrl":"10.1002/cpa.22246","url":null,"abstract":"<p>We consider the global approximate controllability of the two-dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier–Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 7","pages":"1285-1319"},"PeriodicalIF":3.1,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143056364","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}

{"title":"Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation","authors":"Jian Ding,&nbsp;Zherui Fan,&nbsp;Lu-Jing Huang","doi":"10.1002/cpa.22243","DOIUrl":"10.1002/cpa.22243","url":null,"abstract":"&lt;p&gt;In this work, we study the critical long-range percolation (LRP) on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathbb {Z}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where an edge connects &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;annotation&gt;$i$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;annotation&gt;$j$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; independently with probability 1 for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$|i-j|=1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and with probability &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;exp&lt;/mi&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mo&gt;∫&lt;/mo&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mo&gt;∫&lt;/mo&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$1-exp lbrace -beta int _i^{i+1}int _j^{j+1}|u-v|^{-2}{rm d}u{rm d}vrbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for some fixed &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$beta &gt;0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 7","pages":"1251-1284"},"PeriodicalIF":3.1,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143049912","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}

{"title":"Phase transition of parabolic Ginzburg–Landau equation with potentials of high-dimensional wells","authors":"Yuning Liu","doi":"10.1002/cpa.22242","DOIUrl":"10.1002/cpa.22242","url":null,"abstract":"<p>In this work, we study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg–Landau systems with potentials of high-dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math. <b>65</b> (2012), no. 6, 833–888) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non-standard boundary condition for them. These results are valid provided that the initial datum of the system is well-prepared under natural energy assumptions.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 6","pages":"1199-1247"},"PeriodicalIF":3.1,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143055044","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}