Journal de Mathematiques Pures et Appliquees最新文献

{"title":"Stability of Hölder regularity and weighted functional inequalities","authors":"Soobin Cho ,&nbsp;Panki Kim","doi":"10.1016/j.matpur.2025.103836","DOIUrl":"10.1016/j.matpur.2025.103836","url":null,"abstract":"<div><div>We study symmetric Dirichlet forms on metric measure spaces, which may possess both strongly local and pure-jump parts. We introduce a new formulation of a tail condition for jump measures and weighted functional inequalities. Our framework accommodates Dirichlet forms with singular jump measures and those associated with trace processes of mixed-type stable processes. Using these new weighted functional inequalities, we establish stable, equivalent characterizations of Hölder regularity for caloric and harmonic functions. As an application of our main result, we prove the Hölder continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103836"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841746","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":"Concentration and oscillation analysis of positive solutions to semilinear elliptic equations with exponential growth in a disc. I","authors":"Daisuke Naimen","doi":"10.1016/j.matpur.2025.103834","DOIUrl":"10.1016/j.matpur.2025.103834","url":null,"abstract":"<div><div>We establish a series of concentration and oscillation estimates for semilinear elliptic equations with exponential nonlinearity <span><math><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></msup></math></span> in a disc. Especially, we show various new results on the supercritical case <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span> which are left open in the previous works. We begin with the concentration analysis of blow-up solutions by extending the scaling and pointwise techniques developed in the previous studies. A striking result is that we detect an infinite sequence of bubbles. The precise characterization of the limit profile, energy, and location of each bubble is given. Moreover, we arrive at a natural interpretation, the infinite sequence of bubbles causes the infinite oscillation of the solutions. Based on this idea and our concentration estimates, we next carry out the oscillation analysis. The results allow us to prove that the intersection number between blow-up solutions and singular functions diverges to infinity. Applying this, we finally demonstrate infinite oscillations of bifurcation diagrams of supercritical equations. We present the results mentioned above through a series of two papers. The present one is devoted to the former part, that is, the concentration analysis.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103834"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792180","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":"Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities","authors":"Zhentao He,&nbsp;Chao Ji","doi":"10.1016/j.matpur.2025.103837","DOIUrl":"10.1016/j.matpur.2025.103837","url":null,"abstract":"<div><div>In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graphs <span><math><mi>G</mi></math></span> with localized nonlinearities<span><span><span><math><mi>D</mi><mi>u</mi><mo>−</mo><mi>ω</mi><mi>u</mi><mo>=</mo><mi>a</mi><msub><mrow><mi>χ</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo></math></span></span></span> where <span><math><mi>D</mi></math></span> is the Dirac operator on <span><math><mi>G</mi></math></span>, <span><math><mi>u</mi><mo>:</mo><mi>G</mi><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, <span><math><mi>ω</mi><mo>∈</mo><mi>R</mi></math></span>, <span><math><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the characteristic function of the compact core <span><math><mi>K</mi></math></span>, and <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>. First, for <span><math><mn>2</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mn>4</mn></math></span>, we prove the existence of normalized solutions to (NLDE) using a perturbation argument. Then, for <span><math><mi>p</mi><mo>≥</mo><mn>4</mn></math></span>, we present sufficient conditions under which the normalized solutions to (NLDE) exist. Finally, we extend these results to the case <span><math><mi>a</mi><mo>&lt;</mo><mn>0</mn></math></span> and, for all <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>, prove the existence of normalized solutions to (NLDE) when <span><math><mi>λ</mi><mo>=</mo><mo>−</mo><mi>m</mi><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is an eigenvalue of the operator <span><math><mi>D</mi></math></span>. In the Appendix, we study the influence of the parameters <span><math><mi>m</mi><mo>,</mo><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103837"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841747","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":"A gradient flow on control space with rough initial condition","authors":"Paul Gassiat ,&nbsp;Florin Suciu","doi":"10.1016/j.matpur.2025.103833","DOIUrl":"10.1016/j.matpur.2025.103833","url":null,"abstract":"<div><div>We consider the (sub-Riemannian type) control problem of finding a path going from an initial point <em>x</em> to a target point <em>y</em>, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the bracket-generating (Hörmander) condition, so that the classical Chow-Rashevskii theorem guarantees the existence of such a path. One natural way to try to solve this problem is via a gradient flow on control space. However, since the corresponding dynamics may have saddle points, any convergence result must rely on suitable (e.g. random) initialisation. We consider the case when this initialisation is irregular, which is conveniently formulated via Lyons' rough path theory. We show that one advantage of this initialisation is that the saddle points are moved to infinity, while minima remain at a finite distance from the starting point. In the step 2-nilpotent case, we further manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The proof is based on combining ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our study comes from the training of deep Residual Neural Nets, in the regime when the number of trainable parameters per layer is smaller than the dimension of the data vector.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103833"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760742","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":"Relative entropy method for particle approximation of the Landau equation for Maxwellian molecules","authors":"José Antonio Carrillo ,&nbsp;Xuanrui Feng ,&nbsp;Shuchen Guo ,&nbsp;Pierre-Emmanuel Jabin ,&nbsp;Zhenfu Wang","doi":"10.1016/j.matpur.2025.103838","DOIUrl":"10.1016/j.matpur.2025.103838","url":null,"abstract":"<div><div>We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103838"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841748","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":"A general nonlinear characterization of stochastic incompleteness","authors":"Gabriele Grillo ,&nbsp;Kazuhiro Ishige ,&nbsp;Matteo Muratori ,&nbsp;Fabio Punzo","doi":"10.1016/j.matpur.2025.103839","DOIUrl":"10.1016/j.matpur.2025.103839","url":null,"abstract":"<div><div>Stochastic incompleteness of a Riemannian manifold <em>M</em> amounts to the nonconservation of probability for the heat semigroup on <em>M</em>. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to <span><math><mi>Δ</mi><mi>W</mi><mo>=</mo><mi>ψ</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span> for one, hence all, general nonlinearity <em>ψ</em> which is only required to be continuous, nondecreasing, with <span><math><mi>ψ</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mi>ψ</mi><mo>&gt;</mo><mn>0</mn></math></span> in <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span>. Similar statements hold for (sub)solutions that may change sign. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><mi>ϕ</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> with bounded initial data for one, hence all, general nonlinearity <em>ϕ</em> which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103839"},"PeriodicalIF":2.3,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792178","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 long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions","authors":"Deqin Zhou ,&nbsp;Felipe Linares","doi":"10.1016/j.matpur.2025.103792","DOIUrl":"10.1016/j.matpur.2025.103792","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider the long time behavior of the solutions of the coupled Schrödinger-KdV system&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;i&lt;/mi&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;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mi&gt;R&lt;/mi&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&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;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mi&gt;R&lt;/mi&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&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;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&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;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&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;)&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;γ&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msup&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;/msup&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; without any size assumption on the initial data in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In the second one, we consider the parameter vector &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;γ&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&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;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&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;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In this case, we give a ‘‘smallness” criterion involving the product of the parameter −&lt;em&gt;β&lt;/em&gt; and a constant depending on the initial data in &lt;span&gt;&lt;","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103792"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094983","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":"Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity","authors":"Mónica Clapp ,&nbsp;Angela Pistoia ,&nbsp;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":"2026-01-01","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}

{"title":"L2 estimates and existence theorems for the ∂‾ operators in infinite dimensions, II","authors":"Zhouzhe Wang ,&nbsp;Jiayang Yu ,&nbsp;Xu Zhang","doi":"10.1016/j.matpur.2025.103811","DOIUrl":"10.1016/j.matpur.2025.103811","url":null,"abstract":"<div><div>This paper is the second part of our series of works to establish <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> estimates and existence theorems for the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span> operators in infinite dimensions. In this part, we consider the most difficult case, i.e., the underlying space is a general pseudo-convex domain. In order to handle this longstanding unsolved problem, we introduce several new concepts and techniques, which have independent interest and may be applied in other places.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103811"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362037","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 spectral eigenvalue set and Beurling dimension on self-similar measures","authors":"Lu Zheng-Yi","doi":"10.1016/j.matpur.2025.103809","DOIUrl":"10.1016/j.matpur.2025.103809","url":null,"abstract":"<div><div>In this work, we study harmonic analysis in self-similar measures. A set <span><math><mi>A</mi></math></span> is called a <em>spectral eigenvalue set</em> of <em>μ</em> if there exists <span><math><mi>Λ</mi><mo>⊂</mo><mi>R</mi></math></span> such that the family <span><math><mo>{</mo><mi>a</mi><mi>Λ</mi><mo>:</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>}</mo></math></span> are spectra for <em>μ</em>. Given a Hadamard triple <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>L</mi><mo>)</mo></math></span>, Łaba and Wang <span><span>[33]</span></span> proved that the associated self-similar measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>D</mi></mrow></msub></math></span> is spectral. We establish that the set<span><span><span><math><mi>T</mi><mo>=</mo><mo>{</mo><mi>t</mi><mo>∈</mo><mi>Z</mi><mo>:</mo><mo>(</mo><mi>q</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>t</mi><mi>L</mi><mo>)</mo><mtext> forms a Hadamard triple</mtext><mo>}</mo><mo>⊇</mo><mo>{</mo><mi>p</mi><mo>∈</mo><mi>Z</mi><mo>:</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span></span></span> constitutes a spectral eigenvalue set for <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>,</mo><mi>D</mi></mrow></msub></math></span>. Furthermore, we demonstrate that for any prescribed Beurling dimension <span><math><mi>s</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>#</mi><mi>D</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>q</mi></mrow></mfrac><mo>]</mo></math></span>, the corresponding spectra have the cardinality of the continuum. This result provides a complete answer to the question posed by Kong, Li and Wang <span><span>[30]</span></span>. As an application, we characterize the eigenvalue sets for <em>N</em>-Bernoulli convolutions, proving that <span><math><mi>A</mi></math></span> is an eigenvalue set if and only if <span><math><mi>A</mi><mo>⊆</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>T</mi></mrow></mfrac><mi>T</mi></math></span> for some <span><math><mi>T</mi><mo>∈</mo><mi>T</mi></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103809"},"PeriodicalIF":2.3,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145319597","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}