Journal of Differential Equations最新文献

{"title":"Simultaneous uniqueness and numerical inversion for an inverse problem in a coupled diffusion system","authors":"Zhiyuan Li ,&nbsp;Chunlong Sun","doi":"10.1016/j.jde.2025.113827","DOIUrl":"10.1016/j.jde.2025.113827","url":null,"abstract":"<div><div>In this work, we investigate an inverse problem in determining multiple coefficients in a coupled diffusion system arising from the time-domain diffuse optical tomography with fluorescence. We simultaneously recover the distribution of the background absorption coefficient, photon diffusion coefficient, and fluorescence absorption in biological tissue by time-dependent boundary measurements. We build the uniqueness theorem for this multiple coefficients simultaneous inverse problem. After that, the numerical inversions for non-smooth absorption featuring various shaped inclusions are considered.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"452 ","pages":"Article 113827"},"PeriodicalIF":2.3,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268970","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":"To what extent does the consideration of positive total flux influence the dynamics of Keller–Segel-type models?","authors":"Khadijeh Baghaei ,&nbsp;Silvia Frassu ,&nbsp;Yuya Tanaka ,&nbsp;Giuseppe Viglialoro","doi":"10.1016/j.jde.2025.113808","DOIUrl":"10.1016/j.jde.2025.113808","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Since the introduction of the Keller-Segel model in 1970 to describe chemotaxis (the interactions between cell distributions &lt;em&gt;u&lt;/em&gt; and chemical distributions &lt;em&gt;v&lt;/em&gt;), there has been a significant proliferation of research articles exploring various extensions and modifications of this model within the scientific community. From a technical standpoint, the totality of results concerning these variants are characterized by the assumption that the total flux, involving both distributions, of the model under consideration is &lt;em&gt;zero&lt;/em&gt;. This research aims to present a novel perspective by focusing on models with a &lt;em&gt;positive&lt;/em&gt; total flux. Specifically, by employing Robin-type boundary conditions for &lt;em&gt;u&lt;/em&gt; and &lt;em&gt;v&lt;/em&gt;, we seek to gain insights into the interactions between cells and their environment, uncovering important dynamics such as how variations in boundary conditions influence chemotactic behavior. In particular, the choice of the boundary conditions is motivated by real-world phenomena and by the fact that the related analysis reveals some interesting properties of the system.&lt;/div&gt;&lt;div&gt;Mathematically, or &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; we investigate Keller–Segel-type models with positive total flux &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, reading as&lt;span&gt;&lt;span&gt;&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;max&lt;/mtext&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;mtr&gt;&lt;mtd&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;max&lt;/mtext&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;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;on&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;max&lt;/mtext&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; where &lt;span","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"452 ","pages":"Article 113808"},"PeriodicalIF":2.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268973","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":"Variational principles of invariance entropy dimension","authors":"Hu Chen ,&nbsp;Yu Huang ,&nbsp;Xingfu Zhong","doi":"10.1016/j.jde.2025.113819","DOIUrl":"10.1016/j.jde.2025.113819","url":null,"abstract":"<div><div>We introduce two types of invariance entropy dimensions (<em>α</em>-invariance entropies), measure-theoretic invariance entropy dimensions (measure-theoretic <em>α</em>-invariance entropies), and measure-theoretic local invariance entropy dimensions (measure-theoretic local <em>α</em>-invariance entropy) to investigate the complexity of a control system with zero invariance entropy, where <span><math><mi>α</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>. Under some reasonable assumptions, we respectively present two variational principles and inverse variational principles for <em>α</em>-invariance entropies and invariance entropy dimensions, and show that Bowen <em>α</em>-invariance entropy (Bowen invariance entropy dimension) and packing <em>α</em>-invariance entropy (packing invariance entropy dimension) can be determined via the corresponding local entropies of measures.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"453 ","pages":"Article 113819"},"PeriodicalIF":2.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270307","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":"Large scale limit for a dispersion-managed NLS","authors":"Jason Murphy","doi":"10.1016/j.jde.2025.113830","DOIUrl":"10.1016/j.jde.2025.113830","url":null,"abstract":"<div><div>We derive the standard power-type NLS as a scaling limit of the Gabitov–Turitsyn dispersion-managed NLS, using the 2<em>d</em> defocusing, cubic equation as a model case. In particular, we obtain global-in-time scattering solutions to the dispersion-managed NLS for large scale data of arbitrary <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113830"},"PeriodicalIF":2.3,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145262744","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":"Multiple solutions for the nonlinear Schrödinger-Poisson system with a partial confinement","authors":"Liying Shan ,&nbsp;Wei Shuai ,&nbsp;Jianghua Ye","doi":"10.1016/j.jde.2025.113815","DOIUrl":"10.1016/j.jde.2025.113815","url":null,"abstract":"<div><div>We study the following nonlinear Schrödinger-Poisson system with a partial confinement<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mo>(</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>u</mi><mo>+</mo><mi>λ</mi><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>=</mo><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><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>ϕ</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>&gt;</mo><mn>0</mn></math></span> is a parameter. The existence and nonexistence results are established by variational methods, depending on the parameters <em>p</em> and <em>λ</em>. It turns out that <span><math><mi>p</mi><mo>=</mo><mn>3</mn></math></span> is a critical value for the existence of solutions.</div><div>Our results can be viewed as an extension of the results of Ruiz <span><span>[33]</span></span> concerning the nonlinear Schrödinger-Poisson equation with a positive constant potential. However, due to the presence of partial confinement, the Nehari-Pohozaev manifold method is no longer applicable in this paper for <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mn>16</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>]</mo></math></span>. We need to explore the more complicated underlying functional geometry with a different variational approach. Moreover, we also construct saddle type nodal solutions whose nodal domains meet at the origin.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113815"},"PeriodicalIF":2.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145266163","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 a scaled abstract linking theorem with an application to the Schrödinger–Poisson–Slater equation","authors":"Kanishka Perera ,&nbsp;Kaye Silva","doi":"10.1016/j.jde.2025.113824","DOIUrl":"10.1016/j.jde.2025.113824","url":null,"abstract":"<div><div>We prove an abstract linking theorem that can be used to show existence of solutions to various types of variational elliptic equations, including Schrödinger–Poisson–Slater type equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"453 ","pages":"Article 113824"},"PeriodicalIF":2.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145236433","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 existence of weak solutions to a Baer–Nunziato type system","authors":"Martin Kalousek,&nbsp;Šárka Nečasová","doi":"10.1016/j.jde.2025.113804","DOIUrl":"10.1016/j.jde.2025.113804","url":null,"abstract":"<div><div>In this paper, we consider a compressible one velocity Baer–Nunziato type system with dissipation describing the evolution of a mixture of two compressible heat conducting fluids. The complete existence proof for weak solutions to this system was addressed as an open problem in <span><span>[12, Section 5]</span></span>. The purpose of this paper is to prove the global in time existence of weak solutions to the one velocity Baer–Nunziato type system for arbitrary large initial data. The goal is achieved in three steps. Firstly, the given system is transformed into a new one which possesses the “Navier-Stokes-Fourier” structure. Secondly, the new system is solved by an adaptation of the Feireisl–Lions approach for solving the compressible full system applying also the almost compactness property introduced by Vasseur et al. <span><span>[19]</span></span>. Finally, the existence of a weak solution to the original one velocity Baer–Nunziato system is shown using the almost uniqueness property of renormalized solutions to pure transport equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"452 ","pages":"Article 113804"},"PeriodicalIF":2.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268971","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":"Harmonic mappings on Grushin planes","authors":"Tomasz Adamowicz ,&nbsp;Marcin Walicki ,&nbsp;Ben Warhurst","doi":"10.1016/j.jde.2025.113806","DOIUrl":"10.1016/j.jde.2025.113806","url":null,"abstract":"<div><div>We study the Grushin spaces given by Hölder regular vector fields with the Carnot–Carathéodory distance, equipped with the natural weighted <em>n</em>-Lebesgue measure. It turns out that such a measure is <em>n</em>-Ahlfors regular and <em>p</em>-Muckenhoupt for <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span>. We investigate the related 2-Dirichlet energy and the harmonic mappings, and focus our attention on harmonic functions and mappings from domains in a Grushin plane to Euclidean spaces <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, for <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>. Our results enclose the second order Sobolev regularity, the Bochner identity and its consequences for the regularity of harmonic mappings, the Liouville-type theorems and a counterpart of the Lewy theorem. Furthermore, the Korevaar–Schoen energy of mappings on Grushin planes is studied and proven to be equivalent to the 2-Dirichlet energy. The discussion is illustrated by several examples.</div><div>We generalize some geometric and regularity results by Ferrari–Valdinoci <span><span>[23]</span></span> and Franchi–Serapioni <span><span>[30]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"453 ","pages":"Article 113806"},"PeriodicalIF":2.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269783","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":"An application of the Ważewski principle to the characterization of continuous probability distributions","authors":"Mariusz Bieniek","doi":"10.1016/j.jde.2025.113817","DOIUrl":"10.1016/j.jde.2025.113817","url":null,"abstract":"<div><div>We introduce an unexpected application of the Ważewski retract principle to a probabilistic challenge: identifying probability distributions via the regression function of ordered statistical data such as order statistics and record values. In order to prevent repetitive arguments across various models, we examine a wide-ranging category of models known as generalized order statistics. We establish a precise formula for the underlying distribution using a specified regression and a solution to an additional Volterra-Stieltjes integral equation. Moreover, we identify the class of feasible regression functions by determining the necessary and sufficient conditions for a function to qualify as a regression. Our findings are supported by examples that highlight the importance of the stipulated conditions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"453 ","pages":"Article 113817"},"PeriodicalIF":2.3,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270305","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":"Fractional De Giorgi conjecture in dimension 2 via complex-plane methods","authors":"Serena Dipierro ,&nbsp;João Gonçalves da Silva ,&nbsp;Giorgio Poggesi ,&nbsp;Enrico Valdinoci","doi":"10.1016/j.jde.2025.113816","DOIUrl":"10.1016/j.jde.2025.113816","url":null,"abstract":"<div><div>We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the <em>s</em>-harmonic extension of the fractional Laplacian in the half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> introduced by L. Caffarelli and L. Silvestre in 2007.</div><div>We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></math></span> under Neumann boundary conditions. This generalizes the <em>s</em>-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"453 ","pages":"Article 113816"},"PeriodicalIF":2.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270306","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}