Journal of Functional Analysis最新文献

{"title":"The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates","authors":"Yihong Du, Wenjie Ni","doi":"10.1016/j.jfa.2024.110649","DOIUrl":"10.1016/j.jfa.2024.110649","url":null,"abstract":"<div><p>This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. In Part 1 <span><span>[19]</span></span>, we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel <span><math><mi>J</mi><mo>(</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo></math></span> such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an associated semi-wave problem established in <span><span>[15]</span></span>. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying <span><math><mi>J</mi><mo>(</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo><mo>∼</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>β</mi></mrow></msup></math></span> for <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>≫</mo><mn>1</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case <span><span>[18]</span></span> when <em>β</em> crosses the value <span><math><mi>N</mi><mo>+</mo><mn>2</mn></math></span>. More precisely, (a) when <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>]</mo></math></span>, we show that for <span><math><mi>t</mi><mo>≫</mo><mn>1</mn></math></span>, <span><math><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>∼</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>1</mn><mo>/</mo><mo>(</mo><mi>β</mi><mo>−</mo><mi>N</mi><mo>)</mo></mrow></msup></math></span> if <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mi>N</mi><mo>,</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>∼</mo><mi>t</mi><mi>ln</mi><mo>⁡</mo><mi>t</mi></math></span> if <span><math><mi>β</mi><mo>=</mo><mi>N</mi><mo>+</mo><mn>1</mn></math></span>, which is of the same pattern as in dimension one, namely we recover the result in <span><span>[18]</span></span> by letting <span><math><mi>N</mi><mo>=</mo><mn>1</mn></math></span> in the above statements; (b) when <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>N</mi><mo>+</mo><mn>2</mn><mo>]</mo></math></span>, the front has a finite spreading speed <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>β</mi><mo>)</mo></math></span> i","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110649"},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Widom–Sobolev formula for discontinuous matrix-valued symbols","authors":"Leon Bollmann,&nbsp;Peter Müller","doi":"10.1016/j.jfa.2024.110651","DOIUrl":"10.1016/j.jfa.2024.110651","url":null,"abstract":"<div><p>We prove the Widom–Sobolev formula for the asymptotic behaviour of truncated Wiener–Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110651"},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003392/pdfft?md5=8010699661a21471a1a6bd426ef43d6a&pid=1-s2.0-S0022123624003392-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary","authors":"Qingyou He","doi":"10.1016/j.jfa.2024.110643","DOIUrl":"10.1016/j.jfa.2024.110643","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We consider the Cauchy problem of the porous medium type reaction-diffusion equation&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;em&gt;g&lt;/em&gt; is the given monotonic decreasing function with the density critical threshold &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We prove that the pressure &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; tends to the pressure critical threshold &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; at the time decay rate &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. If the initial density &lt;span&gt;&lt;math&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is compactly supported, we justify that the support &lt;span&gt;&lt;math&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of the density &lt;em&gt;ρ&lt;/em&gt; expands exponentially in time. Furthermore, we show that there exists a time &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; such that the pressure &lt;em&gt;P&lt;/em&gt; is Lipschitz continuous for &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, which is the optimal (sharp) regularity of the pressure, and the free surface &lt;span&gt;&lt;math&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110643"},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the optimal rate for the convergence problem in mean field control","authors":"Samuel Daudin ,&nbsp;François Delarue ,&nbsp;Joe Jackson","doi":"10.1016/j.jfa.2024.110660","DOIUrl":"10.1016/j.jfa.2024.110660","url":null,"abstract":"<div><p>The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, with <em>N</em> being the number of particles, and we verify that <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mo>(</mo><mn>3</mn><mi>d</mi><mo>+</mo><mn>6</mn><mo>)</mo></mrow></msup></math></span>. We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span>, and we conjecture that the optimal rate should indeed be exactly <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>d</mi></mrow></msup></math></span> (at least when <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110660"},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003483/pdfft?md5=5935f79205a900c37dc8ad1a51b88e6a&pid=1-s2.0-S0022123624003483-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Norm convergence rate for multivariate quadratic polynomials of Wigner matrices","authors":"Jacob Fronk ,&nbsp;Torben Krüger ,&nbsp;Yuriy Nemish","doi":"10.1016/j.jfa.2024.110647","DOIUrl":"10.1016/j.jfa.2024.110647","url":null,"abstract":"<div><p>We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension <em>N</em> of the matrices grows to infinity, the operator norm of such polynomials <em>q</em> converges to a deterministic limit with a rate of convergence of <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110647"},"PeriodicalIF":1.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003355/pdfft?md5=4e27857829ee38213729e119afe883b6&pid=1-s2.0-S0022123624003355-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Small perturbations of polytopes","authors":"Christian Kipp","doi":"10.1016/j.jfa.2024.110644","DOIUrl":"10.1016/j.jfa.2024.110644","url":null,"abstract":"<div><p>Motivated by first-order conditions for extremal bodies of geometric functionals, we study a functional analytic notion of infinitesimal perturbations of convex bodies and give a full characterization of the set of realizable perturbations if the perturbed body is a polytope. As an application, we derive a necessary condition for polytopal maximizers of the isotropic constant.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110644"},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S002212362400332X/pdfft?md5=e462643d71502f41002932b5ab067bea&pid=1-s2.0-S002212362400332X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Locally universal C⁎-algebras with computable presentations","authors":"Alec Fox ,&nbsp;Isaac Goldbring ,&nbsp;Bradd Hart","doi":"10.1016/j.jfa.2024.110652","DOIUrl":"10.1016/j.jfa.2024.110652","url":null,"abstract":"<div><p>The Kirchberg Embedding Problem (KEP) asks if every <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra embeds into an ultrapower of the Cuntz algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra with a computable presentation.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110652"},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Random Carleson sequences for the Hardy space on the polydisc and the unit ball","authors":"Nikolaos Chalmoukis ,&nbsp;Alberto Dayan ,&nbsp;Giuseppe Lamberti","doi":"10.1016/j.jfa.2024.110659","DOIUrl":"10.1016/j.jfa.2024.110659","url":null,"abstract":"<div><p>We study the Kolmogorov <span><math><mn>0</mn><mo>−</mo><mn>1</mn></math></span> law for a random sequence with prescribed radii so that it generates a Carleson measure almost surely, both for the Hardy space on the polydisc and the Hardy space on the unit ball, thus providing improved versions of previous results of the first two authors and of a separate result of Massaneda. In the polydisc, the geometry of such sequences is not well understood, so we proceed by studying the random Gramians generated by random sequences, using tools from the theory of random matrices. Another result we prove, and that is of its own relevance, is the <span><math><mn>0</mn><mo>−</mo><mn>1</mn></math></span> law for a random sequence to be partitioned into <em>M</em> separated sequences with respect to the pseudo-hyperbolic distance, which is used also to describe the random sequences that are interpolating for the Bloch space on the unit disc almost surely.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 12","pages":"Article 110659"},"PeriodicalIF":1.7,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003471/pdfft?md5=002a3ca282e7fe0e69899ce4b770adcb&pid=1-s2.0-S0022123624003471-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142161637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials","authors":"Esther Cabezas-Rivas ,&nbsp;Salvador Moll ,&nbsp;Marcos Solera","doi":"10.1016/j.jfa.2024.110642","DOIUrl":"10.1016/j.jfa.2024.110642","url":null,"abstract":"<div><p>We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic <em>p</em>-harmonic functions or <em>p</em>-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of <em>p</em>-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of <em>BV</em>-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 11","pages":"Article 110642"},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003306/pdfft?md5=b8aed4f15cb7f0e5872276ff50c4a2cd&pid=1-s2.0-S0022123624003306-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A geometric Elliott invariant and noncommutative rigidity of mapping tori","authors":"Hao Guo ,&nbsp;Valerio Proietti ,&nbsp;Hang Wang","doi":"10.1016/j.jfa.2024.110625","DOIUrl":"10.1016/j.jfa.2024.110625","url":null,"abstract":"<div><p>We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions on a compact space can be lifted to an isomorphism of their foliation <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"287 11","pages":"Article 110625"},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003136/pdfft?md5=a1cbbfe5bdbc4ef488f7c160f8a48b02&pid=1-s2.0-S0022123624003136-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142076533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}