Pub Date : 2025-09-03DOI: 10.1016/j.matpur.2025.103779
Bernd Schmidt, Martin Steinbach
We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.
{"title":"Korn type inequalities for objective structures","authors":"Bernd Schmidt, Martin Steinbach","doi":"10.1016/j.matpur.2025.103779","DOIUrl":"10.1016/j.matpur.2025.103779","url":null,"abstract":"<div><div>We establish discrete Korn type inequalities for particle systems within the general class of objective structures that represents a far reaching generalization of crystal lattice structures. For space filling configurations whose symmetry group is a general space group we obtain a full discrete Korn inequality. For systems with non-trivial codimension our results provide an intrinsic rigidity estimate within the extended dimensions of the structure. As their continuum counterparts in elasticity theory, such estimates are at the core of energy estimates and, hence, a stability analysis for a wide class of atomistic particle systems.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103779"},"PeriodicalIF":2.3,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104685","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}
Pub Date : 2025-08-28DOI: 10.1016/j.matpur.2025.103783
Mónica Clapp , Angela Pistoia , Alberto Saldaña
We consider a nonautonomous semilinear elliptic problem where the power-type nonlinearity is multiplied by a discontinuous coefficient that takes the value one inside a bounded open set Ω and minus one in its complement. In the slightly subcritical regime, we prove the existence of concentrating positive and nodal solutions. Moreover, depending on the geometry of Ω, we establish multiplicity of positive solutions. Finally, in the critical case, we show the existence of a blow-up positive solution when Ω has nontrivial topology. Our proofs rely on a Lyapunov-Schmidt reduction strategy which in these problems turns out to be remarkably simple. We take this opportunity to highlight certain aspects of the method that are often overlooked and present it in a more accessible and detailed manner for nonexperts.
{"title":"Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity","authors":"Mónica Clapp , Angela Pistoia , Alberto Saldaña","doi":"10.1016/j.matpur.2025.103783","DOIUrl":"10.1016/j.matpur.2025.103783","url":null,"abstract":"<div><div>We consider a nonautonomous semilinear elliptic problem where the power-type nonlinearity is multiplied by a discontinuous coefficient that takes the value one inside a bounded open set Ω and minus one in its complement. In the slightly subcritical regime, we prove the existence of concentrating positive and nodal solutions. Moreover, depending on the geometry of Ω, we establish multiplicity of positive solutions. Finally, in the critical case, we show the existence of a blow-up positive solution when Ω has nontrivial topology. Our proofs rely on a Lyapunov-Schmidt reduction strategy which in these problems turns out to be remarkably simple. We take this opportunity to highlight certain aspects of the method that are often overlooked and present it in a more accessible and detailed manner for nonexperts.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103783"},"PeriodicalIF":2.3,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988543","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}
Pub Date : 2025-08-28DOI: 10.1016/j.matpur.2025.103784
Giorgio Poggesi
The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in Ciraolo and Maggi (2017) [3], where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in Julin and Niinikoski (2023) [12], where -deviations are considered for domains in . In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation which is clearly even weaker than .
{"title":"Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with L1-type deviations","authors":"Giorgio Poggesi","doi":"10.1016/j.matpur.2025.103784","DOIUrl":"10.1016/j.matpur.2025.103784","url":null,"abstract":"<div><div>The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in Ciraolo and Maggi (2017) <span><span>[3]</span></span>, where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in Julin and Niinikoski (2023) <span><span>[12]</span></span>, where <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>-deviations are considered for domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation<span><span><span><math><munder><mo>∫</mo><mrow><mo>∂</mo><mi>Ω</mi></mrow></munder><msup><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mtext> where </mtext><mrow><mo>{</mo><mtable><mtr><mtd><mi>H</mi><mtext> is the mean curvature of </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mo>|</mo><mo>∂</mo><mi>Ω</mi><mo>|</mo></mrow><mrow><mi>N</mi><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>:</mo><mo>=</mo><mi>max</mi><mo></mo><mrow><mo>{</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>,</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> which is clearly even weaker than <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></mrow></msub></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103784"},"PeriodicalIF":2.3,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145009927","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}
Pub Date : 2025-08-28DOI: 10.1016/j.matpur.2025.103781
Shuang Liu
The paper is concerned with the effects of the spatio-temporal heterogeneity on the principal eigenvalues of some linear time-periodic parabolic systems. Various asymptotic behaviors of the principal eigenvalue and its monotonicity, as a function of the diffusion rate and frequency, are derived. In particular, some singular behaviors of the principal eigenvalues are characterized when both the diffusion rate and frequency approach zero, with some scalar time-periodic Hamilton-Jacobi equation as the limiting equation. Furthermore, we completely classify the topological structures of the level sets for the principal eigenvalues in the plane of the diffusion rate and frequency. Our results not only generalize the findings in [28] for scalar periodic-parabolic operators, but also reveal more rich global information, for time-periodic parabolic systems, on the dependence of the principal eigenvalues upon the spatio-temporal heterogeneity.
{"title":"Asymptotic and global analysis of principal eigenvalues for linear time-periodic parabolic systems","authors":"Shuang Liu","doi":"10.1016/j.matpur.2025.103781","DOIUrl":"10.1016/j.matpur.2025.103781","url":null,"abstract":"<div><div>The paper is concerned with the effects of the spatio-temporal heterogeneity on the principal eigenvalues of some linear time-periodic parabolic systems. Various asymptotic behaviors of the principal eigenvalue and its monotonicity, as a function of the diffusion rate and frequency, are derived. In particular, some singular behaviors of the principal eigenvalues are characterized when both the diffusion rate and frequency approach zero, with some scalar time-periodic Hamilton-Jacobi equation as the limiting equation. Furthermore, we completely classify the topological structures of the level sets for the principal eigenvalues in the plane of the diffusion rate and frequency. Our results not only generalize the findings in <span><span>[28]</span></span> for scalar periodic-parabolic operators, but also reveal more rich global information, for time-periodic parabolic systems, on the dependence of the principal eigenvalues upon the spatio-temporal heterogeneity.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103781"},"PeriodicalIF":2.3,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144925037","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}
Pub Date : 2025-08-28DOI: 10.1016/j.matpur.2025.103782
Kai Du , Zhenjie Ren , Florin Suciu , Songbo Wang
For a certain class of McKean–Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean–Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron.
{"title":"Self-interacting approximation to McKean–Vlasov long-time limit: A Markov chain Monte Carlo method","authors":"Kai Du , Zhenjie Ren , Florin Suciu , Songbo Wang","doi":"10.1016/j.matpur.2025.103782","DOIUrl":"10.1016/j.matpur.2025.103782","url":null,"abstract":"<div><div>For a certain class of McKean–Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean–Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103782"},"PeriodicalIF":2.3,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094980","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}
Pub Date : 2025-08-28DOI: 10.1016/j.matpur.2025.103780
Salvatore Federico , Fausto Gozzi , Andrzej Święch
We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.
{"title":"On mean field games in infinite dimension","authors":"Salvatore Federico , Fausto Gozzi , Andrzej Święch","doi":"10.1016/j.matpur.2025.103780","DOIUrl":"10.1016/j.matpur.2025.103780","url":null,"abstract":"<div><div>We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman <span><span>(<strong>HJB</strong>)</span></span> equation in the paper, coupled with a nonlinear Fokker-Planck <span><span>(<strong>FP</strong>)</span></span> equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103780"},"PeriodicalIF":2.3,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010528","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}
Pub Date : 2025-07-22DOI: 10.1016/j.matpur.2025.103777
Thérèse Moerschell , Massimo Sorella
We present a novel example of a divergence–free velocity field for arbitrary but fixed which leads to non-unique solutions of advection–diffusion in the class while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence–free velocity field , for fixed, but arbitrary, with non–unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for divergence–free velocity fields with could lead to non–unique solutions in the space
我们给出了一个新的无散度速度场b∈L∞(（0,1);Lp(T2)）对于p<；2的任意但固定的例子，它导致了在满足局部能量不等式的Lt，x∞∩Lt2Hx1类中的平流扩散的非唯一解。该结果补充了无散度和lx2可积速度场有界解的已知唯一性结果。此外，为了得到唯一性结果，我们还证明了速度场时间可积性的必要性。更准确地说，我们构造了另一个无散度速度场b∈Lp((0,1);L∞(T2))，对于p<；2是固定的，但是任意的，具有上述非唯一解。在这方面，我们的贡献缩小了独特性和非独特性制度之间的差距。先前，用凸积分技术证明了对于d≥3个无散度速度场b∈L∞((0,1);Lp(Td)) with p<；2dd+2可以导致空间Lt∞Lx2dd−2∩Lt2Hx1的非唯一解。我们的证明是基于随机拉格朗日方法，而不依赖于凸积分。
{"title":"Non-uniqueness of parabolic solutions for advection-diffusion equation","authors":"Thérèse Moerschell , Massimo Sorella","doi":"10.1016/j.matpur.2025.103777","DOIUrl":"10.1016/j.matpur.2025.103777","url":null,"abstract":"<div><div>We present a novel example of a divergence–free velocity field <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span> for <span><math><mi>p</mi><mo><</mo><mn>2</mn></math></span> arbitrary but fixed which leads to non-unique solutions of advection–diffusion in the class <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>x</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>∩</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>H</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence–free velocity field <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span>, for <span><math><mi>p</mi><mo><</mo><mn>2</mn></math></span> fixed, but arbitrary, with non–unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> divergence–free velocity fields <span><math><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span> with <span><math><mi>p</mi><mo><</mo><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></span> could lead to non–unique solutions in the space <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></msubsup><mo>∩</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msubsup><msubsup><mrow><mi>H</mi></mrow><mrow><","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103777"},"PeriodicalIF":2.3,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738741","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}
Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103772
Timo S. Hänninen , Emiel Lorist , Jaakko Sinko
<div><div>As our main result, we supply the missing characterization of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> boundedness of the commutator of a non-degenerate Calderón–Zygmund operator <em>T</em> and pointwise multiplication by <em>b</em> for exponents <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> and Muckenhoupt weights <span><math><mi>μ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Namely, the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo><mo>:</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> is bounded if and only if <em>b</em> satisfies the following new, cancellative condition:<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mi>q</mi><mo>/</mo><mo>(</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>ν</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi></math></span> is the weighted sharp maximal function defined by<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>:</mo><mo>=</mo><munder><mi>sup</mi><mrow><mi>Q</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mrow><mn>1</mn></mrow><mrow><mi>Q</mi></mrow></msub></mrow><mrow><mi>ν</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mfrac><munder><mo>∫</mo><mrow><mi>Q</mi></mrow></munder><mo>|</mo><mi>b</mi><mo>−</mo><msub><mrow><mo>〈</mo><mi>b</mi><mo>〉</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>|</mo><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> and <em>ν</em> is the Bloom weight defined by <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msup></math></span>.</div><div>In the unweighted case <span><math><mi>μ</mi><mo>=</mo><mi>λ</mi><mo>=</mo><mn>1</mn></math></span>, by a result of Hytönen the boundedness of the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo></math></span> is, after factoring out constants, characterized by the boundedness
作为我们的主要结果，我们提供了对于指数1<；q<p<；∞和Muckenhoupt权μ∈Ap和λ∈Aq的非简并Calderón-Zygmund算子T的对易子的Lp（μ）→Lq（λ）有界性和点向乘b的缺失表征。即，换向子[b，T]:Lp（μ）→Lq（λ）有界当且仅当b满足以下新的可消条件：Mν#b∈Lpq/(p−q)(ν)，其中Mν#b是Mν#b定义的加权极大函数：=supQ (q)∫q |b−< b > q |dx， ν是ν1/p+1/q ':=μ1/pλ−1/q定义的Bloom权值。在μ=λ=1的未加权情况下，由Hytönen的结果可知，对易子[b，T]的有界性，在分解出常数后，表征为点向乘以b的有界性，即b∈Lpq/(p−q)为不可消去条件。我们提供了一个反例，表明在μ∈Ap和λ∈Aq的加权情况下，这种表征被打破。因此，引入新的消去条件是必要的。与对易子并行，我们也刻画了在缺失指数范围p≠q的并矢副积Πb的加权有界性。结合之前在互补指数范围内的结果，我们的结果完成了对所有指数p，q∈(1,∞)的对易子和副积的加权有界性的刻画。
{"title":"Weighted Lp → Lq-boundedness of commutators and paraproducts in the Bloom setting","authors":"Timo S. Hänninen , Emiel Lorist , Jaakko Sinko","doi":"10.1016/j.matpur.2025.103772","DOIUrl":"10.1016/j.matpur.2025.103772","url":null,"abstract":"<div><div>As our main result, we supply the missing characterization of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> boundedness of the commutator of a non-degenerate Calderón–Zygmund operator <em>T</em> and pointwise multiplication by <em>b</em> for exponents <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> and Muckenhoupt weights <span><math><mi>μ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> and <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Namely, the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo><mo>:</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>μ</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>λ</mi><mo>)</mo></math></span> is bounded if and only if <em>b</em> satisfies the following new, cancellative condition:<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mi>q</mi><mo>/</mo><mo>(</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>ν</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi></math></span> is the weighted sharp maximal function defined by<span><span><span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>ν</mi></mrow><mrow><mi>#</mi></mrow></msubsup><mi>b</mi><mo>:</mo><mo>=</mo><munder><mi>sup</mi><mrow><mi>Q</mi></mrow></munder><mo></mo><mfrac><mrow><msub><mrow><mn>1</mn></mrow><mrow><mi>Q</mi></mrow></msub></mrow><mrow><mi>ν</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mfrac><munder><mo>∫</mo><mrow><mi>Q</mi></mrow></munder><mo>|</mo><mi>b</mi><mo>−</mo><msub><mrow><mo>〈</mo><mi>b</mi><mo>〉</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>|</mo><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> and <em>ν</em> is the Bloom weight defined by <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>μ</mi></mrow><mrow><mn>1</mn><mo>/</mo><mi>p</mi></mrow></msup><msup><mrow><mi>λ</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msup></math></span>.</div><div>In the unweighted case <span><math><mi>μ</mi><mo>=</mo><mi>λ</mi><mo>=</mo><mn>1</mn></math></span>, by a result of Hytönen the boundedness of the commutator <span><math><mo>[</mo><mi>b</mi><mo>,</mo><mi>T</mi><mo>]</mo></math></span> is, after factoring out constants, characterized by the boundedness","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103772"},"PeriodicalIF":2.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860693","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}
Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103773
Anatole Gallouët , Quentin Mérigot , Boris Thibert
The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +∞ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.
{"title":"Strong c-concavity and stability in optimal transport","authors":"Anatole Gallouët , Quentin Mérigot , Boris Thibert","doi":"10.1016/j.matpur.2025.103773","DOIUrl":"10.1016/j.matpur.2025.103773","url":null,"abstract":"<div><div>The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong <em>c</em>-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions <em>c</em>. We then introduce a differential criterion for proving that a function is strongly <em>c</em>-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +∞ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103773"},"PeriodicalIF":2.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739250","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}
Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103776
Margherita Solci
We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order . When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for k-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant times the k-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford–Shah functional can be obtained by letting k tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.
{"title":"Local interpolation techniques for higher-order singular perturbations of non-convex functionals: Free-discontinuity problems","authors":"Margherita Solci","doi":"10.1016/j.matpur.2025.103776","DOIUrl":"10.1016/j.matpur.2025.103776","url":null,"abstract":"<div><div>We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for <em>k</em>-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> times the <em>k</em>-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford–Shah functional can be obtained by letting <em>k</em> tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103776"},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702622","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}
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