Communications on Pure and Applied Mathematics最新文献

{"title":"Semiconvexity estimates for nonlinear integro-differential equations","authors":"Xavier Ros-Oton,&nbsp;Clara Torres-Latorre,&nbsp;Marvin Weidner","doi":"10.1002/cpa.22237","DOIUrl":"10.1002/cpa.22237","url":null,"abstract":"<p>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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 3","pages":"592-647"},"PeriodicalIF":3.1,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22237","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642581","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":"Rectifiability, finite Hausdorff measure, and compactness for non-minimizing Bernoulli free boundaries","authors":"Dennis Kriventsov,&nbsp;Georg S. Weiss","doi":"10.1002/cpa.22226","DOIUrl":"10.1002/cpa.22226","url":null,"abstract":"&lt;p&gt;While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about &lt;i&gt;critical points&lt;/i&gt; of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time-dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;annotation&gt;$u$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;—which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Delta u$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a &lt;i&gt;frequency formula&lt;/i&gt; for the Bernoulli problem as well as the celebrated &lt;i&gt;Naber–Valtorta procedure&lt;/i&gt; to answer this more than 20 year old question in an affirmative way: For a closed class we call &lt;i&gt;variational solutions&lt;/i&gt; of the Bernoulli problem, we show that the topological free boundary &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial lbrace u &gt; 0rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (including &lt;i&gt;degenerate&lt;/i&gt; singular points &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;annotation&gt;$x$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, at which &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;mo&gt;·&lt;/mo&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$u(x + r cdot)/r rightarrow 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$rrightarrow 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) is countably &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 3","pages":"545-591"},"PeriodicalIF":3.1,"publicationDate":"2024-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142439720","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":"On the Pólya conjecture for the Neumann problem in planar convex domains","authors":"N. Filonov","doi":"10.1002/cpa.22231","DOIUrl":"10.1002/cpa.22231","url":null,"abstract":"&lt;p&gt;Denote by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N_{cal N} (Omega,lambda)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the counting function of the spectrum of the Neumann problem in the domain &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;annotation&gt;$Omega$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on the plane. G. Pólya conjectured that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N_{cal N} (Omega,lambda) geqslant (4pi)^{-1} |Omega | lambda$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We prove that for convex domains &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Ω&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;msqrt&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msqrt&gt;\u0000 &lt;mspace&gt;&lt;/mspace&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;j&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 3","pages":"537-544"},"PeriodicalIF":3.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142405396","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":"Smooth asymptotics for collapsing Calabi–Yau metrics","authors":"Hans-Joachim Hein,&nbsp;Valentino Tosatti","doi":"10.1002/cpa.22228","DOIUrl":"10.1002/cpa.22228","url":null,"abstract":"<p>We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mtext>th</mtext>\u0000 </mrow>\u0000 <annotation>$ktext{th}$</annotation>\u0000 </semantics></math>-order remainders that satisfy uniform <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mi>k</mi>\u0000 </msup>\u0000 <annotation>$C^k$</annotation>\u0000 </semantics></math>-estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$k = 0$</annotation>\u0000 </semantics></math> known from previous work of the second-named author. For <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$k &gt; 0$</annotation>\u0000 </semantics></math>, the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 2","pages":"382-499"},"PeriodicalIF":3.1,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142397713","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":"Uniqueness of the blow-down limit for a triple junction problem","authors":"Zhiyuan Geng","doi":"10.1002/cpa.22230","DOIUrl":"10.1002/cpa.22230","url":null,"abstract":"<p>We prove the uniqueness of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$L^1$</annotation>\u0000 </semantics></math> blow-down limit at infinity for an entire minimizing solution <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 <mo>:</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$u:mathbb {R}^2rightarrow mathbb {R}^2$</annotation>\u0000 </semantics></math> of a planar Allen–Cahn system with a triple-well potential. Consequently, <span></span><math>\u0000 <semantics>\u0000 <mi>u</mi>\u0000 <annotation>$u$</annotation>\u0000 </semantics></math> can be approximated by a triple junction map at infinity. The proof exploits a careful analysis of energy upper and lower bounds, ensuring that the diffuse interface remains within a small neighborhood of the approximated triple junction at all scales.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 2","pages":"500-534"},"PeriodicalIF":3.1,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142397712","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":"Integral formulation of Klein–Gordon singular waveguides","authors":"Guillaume Bal,&nbsp;Jeremy Hoskins,&nbsp;Solomon Quinn,&nbsp;Manas Rachh","doi":"10.1002/cpa.22227","DOIUrl":"10.1002/cpa.22227","url":null,"abstract":"<p>We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 2","pages":"323-365"},"PeriodicalIF":3.1,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22227","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384201","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":"On minimizers in the liquid drop model","authors":"Otis Chodosh,&nbsp;Ian Ruohoniemi","doi":"10.1002/cpa.22229","DOIUrl":"10.1002/cpa.22229","url":null,"abstract":"<p>We prove that round balls of volume <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>≤</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$le 1$</annotation>\u0000 </semantics></math> uniquely minimize in Gamow's liquid drop model.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 2","pages":"366-381"},"PeriodicalIF":3.1,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384202","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 wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates","authors":"Yu Deng,&nbsp;Alexandru D. Ionescu,&nbsp;Fabio Pusateri","doi":"10.1002/cpa.22224","DOIUrl":"10.1002/cpa.22224","url":null,"abstract":"&lt;p&gt;Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV-type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one-dimensional interface, which is a large torus &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;${mathbb {T}}_R$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of size &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Rge 1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$L^infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-type norm is small. This energy inequality is of “quintic” type: if the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$L^infty$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; norm is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$O(varepsilon)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then the increment of the high-order energies is controlled for times of the order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$varepsilon ^{-3}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 2","pages":"211-322"},"PeriodicalIF":3.1,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142170872","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":"Convergence to the planar interface for a nonlocal free-boundary evolution","authors":"Felix Otto,&nbsp;Richard Schubert,&nbsp;Maria G. Westdickenberg","doi":"10.1002/cpa.22225","DOIUrl":"10.1002/cpa.22225","url":null,"abstract":"<p>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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"161-208"},"PeriodicalIF":3.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142410","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":"Asymptotics of block Toeplitz determinants with piecewise continuous symbols","authors":"Estelle Basor,&nbsp;Torsten Ehrhardt,&nbsp;Jani A. Virtanen","doi":"10.1002/cpa.22223","DOIUrl":"10.1002/cpa.22223","url":null,"abstract":"<p>We determine the asymptotics of the block Toeplitz determinants <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>det</mo>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ϕ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$det T_n(phi)$</annotation>\u0000 </semantics></math> as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$nrightarrow infty$</annotation>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>×</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$Ntimes N$</annotation>\u0000 </semantics></math> matrix-valued piecewise continuous functions <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math> with a finitely many jumps under mild additional conditions. In particular, we prove that\u0000\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"120-160"},"PeriodicalIF":3.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090102","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}