Pub Date : 2025-10-07DOI: 10.1016/j.na.2025.113973
Gabriele Fioravanti
In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem where , , , and is the lower dimensional manifold where the equation loses uniform ellipticity.
Our primary objective is to establish and regularity estimates up to , under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a blow-up argument.
{"title":"The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains","authors":"Gabriele Fioravanti","doi":"10.1016/j.na.2025.113973","DOIUrl":"10.1016/j.na.2025.113973","url":null,"abstract":"<div><div>In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem <span><math><mfenced><mrow><mtable><mtr><mtd><mo>−</mo><mo>div</mo><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow></mrow><mrow><mi>a</mi></mrow></msup><mi>A</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow></mrow><mrow><mi>a</mi></mrow></msup><mi>f</mi><mo>+</mo><mo>div</mo><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow></mrow><mrow><mi>a</mi></mrow></msup><mi>F</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>ψ</mi><mo>,</mo><mspace></mspace><mtext>on</mtext><msub><mrow><mi>Σ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace></mspace></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mi>d</mi></mrow></math></span>, <span><math><mrow><mi>a</mi><mo>+</mo><mi>n</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>Σ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>|</mo><mi>y</mi><mo>|</mo></mrow><mo>=</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span> is the lower dimensional manifold where the equation loses uniform ellipticity.</div><div>Our primary objective is to establish <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span> regularity estimates up to <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a blow-up argument.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113973"},"PeriodicalIF":1.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270054","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}
Pub Date : 2025-10-07DOI: 10.1016/j.na.2025.113977
Ştefana-Lucia Aniţa
This paper concerns a Mean Field Game (MFG) system related to a Nash type equilibrium for dynamical games associated to large populations. One shows that the MFG system may be viewed as the Euler–Lagrange system for an optimal control problem related to a Fokker–Planck equation with control in the drift. One derives the existence of a weak solution to the MFG system and under more restrictive assumptions one proves some uniqueness results.
{"title":"A Mean Field Game system and a related deterministic optimal control problem","authors":"Ştefana-Lucia Aniţa","doi":"10.1016/j.na.2025.113977","DOIUrl":"10.1016/j.na.2025.113977","url":null,"abstract":"<div><div>This paper concerns a Mean Field Game (MFG) system related to a Nash type equilibrium for dynamical games associated to large populations. One shows that the MFG system may be viewed as the Euler–Lagrange system for an optimal control problem related to a Fokker–Planck equation with control in the drift. One derives the existence of a weak solution to the MFG system and under more restrictive assumptions one proves some uniqueness results.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113977"},"PeriodicalIF":1.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270056","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}
Pub Date : 2025-10-07DOI: 10.1016/j.na.2025.113969
Jie Cheng , Tianrui Bai , Fangqi Chen
In this paper, we consider the Riemann problem of the Aw–Rascle traffic model with a damping term and the formation of delta shock waves in the limit of the Riemann solutions as . By introducing a new variable and employing generalized characteristic analysis methods, we construct solutions to the Riemann problem of the inhomogeneous Aw–Rascle traffic model. Specially, for the case , we prove the existence of a critical value for such that when , the Riemann solutions contain no vacuum states; otherwise, a vacuum state emerges. Furthermore, we demonstrate that as , the limit of the Riemann solutions with vacuum states aligns with the Riemann solutions to the inhomogeneous transport model under the same initial conditions, while the limit of solutions with shock waves converges to a curved delta shock solution. Notably, the weights supported on the delta shock solution differ from the Riemann solutions to the inhomogeneous transport model due to the influence of the damping term.
{"title":"Formation of delta shock waves in the limit of Riemann solutions to the Aw–Rascle traffic model with a damping term","authors":"Jie Cheng , Tianrui Bai , Fangqi Chen","doi":"10.1016/j.na.2025.113969","DOIUrl":"10.1016/j.na.2025.113969","url":null,"abstract":"<div><div>In this paper, we consider the Riemann problem of the Aw–Rascle traffic model with a damping term and the formation of delta shock waves in the limit of the Riemann solutions as <span><math><mrow><mi>γ</mi><mo>→</mo><mn>1</mn></mrow></math></span>. By introducing a new variable and employing generalized characteristic analysis methods, we construct solutions to the Riemann problem of the inhomogeneous Aw–Rascle traffic model. Specially, for the case <span><math><mrow><mn>0</mn><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>−</mo></mrow></msub><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mo>+</mo></mrow></msub></mrow></math></span>, we prove the existence of a critical value <span><math><msub><mrow><mover><mrow><mi>γ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>0</mn></mrow></msub></math></span> for <span><math><mi>γ</mi></math></span> such that when <span><math><mrow><mn>0</mn><mo><</mo><mi>γ</mi><mo><</mo><msub><mrow><mover><mrow><mi>γ</mi></mrow><mo>¯</mo></mover></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span>, the Riemann solutions contain no vacuum states; otherwise, a vacuum state emerges. Furthermore, we demonstrate that as <span><math><mrow><mi>γ</mi><mo>→</mo><mn>1</mn></mrow></math></span>, the limit of the Riemann solutions with vacuum states aligns with the Riemann solutions to the inhomogeneous transport model under the same initial conditions, while the limit of solutions with shock waves converges to a curved delta shock solution. Notably, the weights supported on the delta shock solution differ from the Riemann solutions to the inhomogeneous transport model due to the influence of the damping term.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113969"},"PeriodicalIF":1.3,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145270055","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}
Pub Date : 2025-10-03DOI: 10.1016/j.na.2025.113967
Giulia Meglioli , Francescantonio Oliva , Francesco Petitta
We show a global existence result for a doubly nonlinear porous medium type equation of the form on a complete and non-compact Riemannian manifold of infinite volume. Here, for , we assume , and . In particular, under the assumptions that supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided and the initial datum is small enough; namely, we establish an explicit bound on the norm of the solution at all positive times, in terms of the norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in , we can establish the same result in the larger interval, i.e. . This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that is non-compact and has infinite measure.
在无限体积的完全非紧黎曼流形M上，给出了形式为ut=Δpum+uq的双非线性多孔介质型方程的整体存在性结果。这里，对于1<；p<；N，我们假设m(p−1)≥1,m>；1和q>；m（p−1）。特别地，在M支持Sobolev不等式的假设下，我们证明了在q>； M (p−1)+pN且初始基准足够小的情况下，该问题的解在时间上全局存在；也就是说，我们根据数据的L1范数，在所有正时刻的解的L∞范数上建立一个显式的界。在附加的假设下，一个poincar型不等式在M中也成立，我们可以在更大的区间，即q>； M （p−1）中建立同样的结果。这个结果没有欧几里得对应物，因为它完全不同于有界欧几里得定义域的情况，因为M是非紧致的并且具有无限的度量。
{"title":"Global existence for a Leibenson type equation with reaction on Riemannian manifolds","authors":"Giulia Meglioli , Francescantonio Oliva , Francesco Petitta","doi":"10.1016/j.na.2025.113967","DOIUrl":"10.1016/j.na.2025.113967","url":null,"abstract":"<div><div>We show a global existence result for a doubly nonlinear porous medium type equation of the form <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mspace></mspace><msup><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></math></span> on a complete and non-compact Riemannian manifold <span><math><mi>M</mi></math></span> of infinite volume. Here, for <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></math></span>, we assume <span><math><mrow><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>m</mi><mo>></mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. In particular, under the assumptions that <span><math><mi>M</mi></math></span> supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>N</mi></mrow></mfrac></mrow></math></span> and the initial datum is small enough; namely, we establish an explicit bound on the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> norm of the solution at all positive times, in terms of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in <span><math><mi>M</mi></math></span>, we can establish the same result in the larger interval, i.e. <span><math><mrow><mi>q</mi><mo>></mo><mi>m</mi><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that <span><math><mi>M</mi></math></span> is non-compact and has infinite measure.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113967"},"PeriodicalIF":1.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222823","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}
Pub Date : 2025-10-03DOI: 10.1016/j.na.2025.113964
Alysson Cunha
We prove that the Benjamin–Ono equation is globally well-posed in for . Our approach does not rely on the global gauge transformation introduced by Tao (Tao, 2004). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin–Ono (DGBO) equation.
{"title":"Improvement of the parabolic regularization method and applications to dispersive models","authors":"Alysson Cunha","doi":"10.1016/j.na.2025.113964","DOIUrl":"10.1016/j.na.2025.113964","url":null,"abstract":"<div><div>We prove that the Benjamin–Ono equation is globally well-posed in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Our approach does not rely on the global gauge transformation introduced by Tao (Tao, 2004). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin–Ono (DGBO) equation.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113964"},"PeriodicalIF":1.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223263","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}
Pub Date : 2025-10-03DOI: 10.1016/j.na.2025.113966
Shu Wang, Rulv Li
We in this paper study the singularity formation and global well-posedness of a nonlocal model for some initial boundary condition with a real parameter, which is a one dimensional weak advection model for the three dimensional incompressible Navier–Stokes equations. Based on the Lyapunov functional and contradiction argument, we can prove that the inviscid nonlocal model develops a finite time blowup solution with some even initial data. But, for some special positive parameter and initial data with the given symbol, the inviscid model also has a global smooth solution by the characteristic’ method. Furthermore, by the energy estimations and Gagliardo–Nirenberg inequality, we also obtain that the viscous nonlocal model has a unique global solution with some initial data with the given symbol for all nonnegative parameter. More specially, there is a particular model to the nonlocal model such that the global solution to this model exists for some negative parameter.
{"title":"Global and singular solution to a nonlocal model of three-dimensional incompressible Navier–Stokes equations","authors":"Shu Wang, Rulv Li","doi":"10.1016/j.na.2025.113966","DOIUrl":"10.1016/j.na.2025.113966","url":null,"abstract":"<div><div>We in this paper study the singularity formation and global well-posedness of a nonlocal model for some initial boundary condition with a real parameter, which is a one dimensional weak advection model for the three dimensional incompressible Navier–Stokes equations. Based on the Lyapunov functional and contradiction argument, we can prove that the inviscid nonlocal model develops a finite time blowup solution with some even initial data. But, for some special positive parameter and initial data with the given symbol, the inviscid model also has a global smooth solution by the characteristic’ method. Furthermore, by the energy estimations and Gagliardo–Nirenberg inequality, we also obtain that the viscous nonlocal model has a unique global solution with some initial data with the given symbol for all nonnegative parameter. More specially, there is a particular model to the nonlocal model such that the global solution to this model exists for some negative parameter.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113966"},"PeriodicalIF":1.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223264","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}
Pub Date : 2025-10-03DOI: 10.1016/j.na.2025.113953
Noemi David , Filippo Santambrogio , Markus Schmidtchen
We present regularity results for nonlinear drift–diffusion equations of porous medium type (together with their incompressible limit). We relax the assumptions imposed on the drift term with respect to previous results and additionally study the effect of linear diffusion on our regularity result (a scenario of particular interest in the incompressible case, for it represents the motion of particles driven by a Brownian motion subject to a density constraint). Specifically, this work concerns the -summability of the pressure gradient in porous medium flows with drifts that is stable with respect to the exponent of the nonlinearity, and -estimates on the pressure Hessian (in particular, in the incompressible case with linear diffusion we prove that the pressure is the positive part of an -function).
{"title":"Uniform regularity estimates for nonlinear diffusion–advection equations in the hard-congestion limit","authors":"Noemi David , Filippo Santambrogio , Markus Schmidtchen","doi":"10.1016/j.na.2025.113953","DOIUrl":"10.1016/j.na.2025.113953","url":null,"abstract":"<div><div>We present regularity results for nonlinear drift–diffusion equations of porous medium type (together with their incompressible limit). We relax the assumptions imposed on the drift term with respect to previous results and additionally study the effect of linear diffusion on our regularity result (a scenario of particular interest in the incompressible case, for it represents the motion of particles driven by a Brownian motion subject to a density constraint). Specifically, this work concerns the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>-summability of the pressure gradient in porous medium flows with drifts that is stable with respect to the exponent of the nonlinearity, and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-estimates on the pressure Hessian (in particular, in the incompressible case with linear diffusion we prove that the pressure is the positive part of an <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-function).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113953"},"PeriodicalIF":1.3,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223262","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}
Pub Date : 2025-09-26DOI: 10.1016/j.na.2025.113952
Hans-Bert Rademacher
We show that for a generic Riemannian metric on a compact manifold of dimension all geodesic loops based at a fixed point have no self-intersections. We also show that for an open and dense subset of the space of Riemannian metrics on an -disc with and with a strictly convex boundary there are geometrically distinct orthogonal geodesic chords without self-intersections. We use a perturbation result for intersecting geodesic segments of the author Rademacher (2024) and a genericity statement due to Bettiol and Giambò (2010) and existence results for orthogonal geodesic chords by Giambò et al. (2018).
{"title":"Geodesic loops and orthogonal geodesic chords without self-intersections","authors":"Hans-Bert Rademacher","doi":"10.1016/j.na.2025.113952","DOIUrl":"10.1016/j.na.2025.113952","url":null,"abstract":"<div><div>We show that for a generic Riemannian metric on a compact manifold of dimension <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> all geodesic loops based at a fixed point have no self-intersections. We also show that for an open and dense subset of the space of Riemannian metrics on an <span><math><mi>n</mi></math></span>-disc with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and with a strictly convex boundary there are <span><math><mi>n</mi></math></span> geometrically distinct orthogonal geodesic chords without self-intersections. We use a perturbation result for intersecting geodesic segments of the author Rademacher (2024) and a genericity statement due to Bettiol and Giambò (2010) and existence results for orthogonal geodesic chords by Giambò et al. (2018).</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113952"},"PeriodicalIF":1.3,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145160065","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}
Pub Date : 2025-09-25DOI: 10.1016/j.na.2025.113948
Francesca Bianchi , Giorgio Stefani , Anna Chiara Zagati
We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the 1-dimensional case, even for non-convex sets, some of which optimal in the case .
{"title":"A geometrical approach to the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces","authors":"Francesca Bianchi , Giorgio Stefani , Anna Chiara Zagati","doi":"10.1016/j.na.2025.113948","DOIUrl":"10.1016/j.na.2025.113948","url":null,"abstract":"<div><div>We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case <span><math><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></math></span> as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the 1-dimensional case, even for non-convex sets, some of which optimal in the case <span><math><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113948"},"PeriodicalIF":1.3,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145134829","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}
Pub Date : 2025-09-25DOI: 10.1016/j.na.2025.113950
Jongmin Han, Kyungwoo Song
We construct multiple solutions of the generalized self-dual abelian Chern–Simons–Higgs equation in a two-dimensional flat torus by the topological degree method.
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