Pub Date : 2026-02-09DOI: 10.1016/j.matpur.2026.103874
Shikun Cui, Lili Wang, Wendong Wang
As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than .
{"title":"Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow","authors":"Shikun Cui, Lili Wang, Wendong Wang","doi":"10.1016/j.matpur.2026.103874","DOIUrl":"10.1016/j.matpur.2026.103874","url":null,"abstract":"<div><div>As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow <span><math><mo>(</mo><mi>A</mi><mi>y</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span> in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than <span><math><mfrac><mrow><mn>24</mn></mrow><mrow><mn>5</mn></mrow></mfrac><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103874"},"PeriodicalIF":2.3,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147428","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 : 2026-02-06DOI: 10.1016/j.matpur.2026.103868
Nicolas Beuvin , Alberto Farina , Berardino Sciunzi
We consider positive solutions, possibly unbounded, to the semilinear equation on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for u, when f is a (locally or globally) Lipschitz-continuous function satisfying . As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of , and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph Ω. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains.
{"title":"Monotonicity for solutions to semilinear problems in epigraphs","authors":"Nicolas Beuvin , Alberto Farina , Berardino Sciunzi","doi":"10.1016/j.matpur.2026.103868","DOIUrl":"10.1016/j.matpur.2026.103868","url":null,"abstract":"<div><div>We consider positive solutions, possibly unbounded, to the semilinear equation <span><math><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for <em>u</em>, when <em>f</em> is a (locally or globally) Lipschitz-continuous function satisfying <span><math><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>≥</mo><mn>0</mn></math></span>. As an application of our new monotonicity theorems, we prove some classification and/or non-existence results. To prove our results, we first establish some new comparison principles for semilinear problems on general unbounded open sets of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, and then we use them to start and to complete a modified version of the moving plane method adapted to the geometry of the epigraph Ω. As a by-product of our analysis, we also prove some new results of uniqueness and symmetry for solutions (possibly unbounded and sign-changing) to the homogeneous Dirichlet BVP for the semilinear Poisson equation in fairly general unbounded domains.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103868"},"PeriodicalIF":2.3,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147427","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 : 2026-01-05DOI: 10.1016/j.matpur.2025.103840
Serena Dipierro , Enrico Valdinoci , Riccardo Villa
We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem.
We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter.
In addition, we also discuss stickiness phenomena for non-local almost minimal sets.
{"title":"On non-local almost minimal sets and an application to the non-local Massari's Problem","authors":"Serena Dipierro , Enrico Valdinoci , Riccardo Villa","doi":"10.1016/j.matpur.2025.103840","DOIUrl":"10.1016/j.matpur.2025.103840","url":null,"abstract":"<div><div>We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem.</div><div>We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter.</div><div>In addition, we also discuss stickiness phenomena for non-local almost minimal sets.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"208 ","pages":"Article 103840"},"PeriodicalIF":2.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948052","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 : 2026-01-05DOI: 10.1016/j.matpur.2025.103843
Li-Jun Du , Wan-Tong Li , Ming-Zhen Xin
In this paper, we study the monostable pulsating fronts for multi-dimensional reaction-diffusion-advection cooperative systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of the spreading speed (Du et al., 2022 [4]). In the present work, we investigate the uniqueness and stability of monostable pulsating fronts with nonzero speed. We first derive some asymptotic behaviors of these fronts as they approach the unstable limiting state. Utilizing these properties, we then prove the uniqueness modulo translation of pulsating fronts with nonzero speed. Finally, we prove that these pulsating fronts are globally asymptotically stable for solutions of the Cauchy problem with front-like initial data. In particular, we establish the uniqueness and global stability of the critical pulsating front. These results are subsequently applied to a two-species competition system.
本文研究了周期介质中多维反应-扩散-平流协同系统的单稳定脉动锋。最近的结果已经解决了脉动锋的存在和传播速度的线性确定性（Du et al., 2022[4]）。本文研究了非零速度单稳态脉冲锋的唯一性和稳定性。我们首先推导出这些前沿逼近不稳定极限状态时的一些渐近行为。利用这些性质，我们证明了非零速度脉动前模平移的唯一性。最后，我们证明了这些脉冲锋对于柯西问题的解是全局渐近稳定的。特别地，我们建立了临界脉冲锋的唯一性和全局稳定性。这些结果随后被应用于两物种竞争系统。
{"title":"Uniqueness and stability of monostable pulsating fronts for multi-dimensional reaction-diffusion-advection systems in periodic media","authors":"Li-Jun Du , Wan-Tong Li , Ming-Zhen Xin","doi":"10.1016/j.matpur.2025.103843","DOIUrl":"10.1016/j.matpur.2025.103843","url":null,"abstract":"<div><div>In this paper, we study the monostable pulsating fronts for multi-dimensional reaction-diffusion-advection cooperative systems in periodic media. Recent results have addressed the existence of pulsating fronts and the linear determinacy of the spreading speed (Du et al., 2022 <span><span>[4]</span></span>). In the present work, we investigate the uniqueness and stability of monostable pulsating fronts with nonzero speed. We first derive some asymptotic behaviors of these fronts as they approach the unstable limiting state. Utilizing these properties, we then prove the uniqueness modulo translation of pulsating fronts with nonzero speed. Finally, we prove that these pulsating fronts are globally asymptotically stable for solutions of the Cauchy problem with front-like initial data. In particular, we establish the uniqueness and global stability of the critical pulsating front. These results are subsequently applied to a two-species competition system.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"208 ","pages":"Article 103843"},"PeriodicalIF":2.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981768","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 : 2026-01-05DOI: 10.1016/j.matpur.2025.103846
Kien Huu Nguyen
In 2006, Budur, Mustaţǎ and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. By the strong monodromy conjecture, it should have a corresponding picture on the arithmetic side of ideals in polynomial rings. In this paper, we try to address this problem. By using an idea inspired by the Hardy-Littlewood circle method, we introduce the notions of abstract exponential sums modulo and motivic oscillation index of an arbitrary ideal of polynomial rings over number fields. In the arithmetic picture, abstract exponential sums modulo and the motivic oscillation index of an ideal should play the role of the Bernstein-Sato polynomial of the corresponding scheme and its maximal non-trivial root respectively. We will provide some properties of the motivic oscillation index of ideals in this paper. On the other hand, based on Igusa's conjecture for exponential sums, we formulate an averaged Igusa conjecture for exponential sums of ideals. In particular, this conjecture and the motivic oscillation index of ideals will have many interesting applications. We will introduce these applications and prove a variant of this conjecture.
{"title":"Exponential sums and motivic oscillation index of arbitrary ideals and their applications","authors":"Kien Huu Nguyen","doi":"10.1016/j.matpur.2025.103846","DOIUrl":"10.1016/j.matpur.2025.103846","url":null,"abstract":"<div><div>In 2006, Budur, Mustaţǎ and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. By the strong monodromy conjecture, it should have a corresponding picture on the arithmetic side of ideals in polynomial rings. In this paper, we try to address this problem. By using an idea inspired by the Hardy-Littlewood circle method, we introduce the notions of abstract exponential sums modulo <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and motivic oscillation index of an arbitrary ideal of polynomial rings over number fields. In the arithmetic picture, abstract exponential sums modulo <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and the motivic oscillation index of an ideal should play the role of the Bernstein-Sato polynomial of the corresponding scheme and its maximal non-trivial root respectively. We will provide some properties of the motivic oscillation index of ideals in this paper. On the other hand, based on Igusa's conjecture for exponential sums, we formulate an averaged Igusa conjecture for exponential sums of ideals. In particular, this conjecture and the motivic oscillation index of ideals will have many interesting applications. We will introduce these applications and prove a variant of this conjecture.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"207 ","pages":"Article 103846"},"PeriodicalIF":2.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979744","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 : 2026-01-05DOI: 10.1016/j.matpur.2025.103845
Xing Liang , Hongze Wang , Qi Zhou
We study the one-dimensional non-self-adjoint Jacobi operators in the almost-periodic media which are “far from” self-adjoint ones. By relaxing the arithmetic condition of the frequency, and regularity conditions of the coefficients, we show the ground states of the operator persists. This result can be seen as complementary of Kozlov's classical result. Besides that, we give two applications: the first is to show the existence and uniqueness of the positive steady state of the discrete Fisher-KPP type equation; the second is to investigate the asymptotic behavior of the discrete stationary parabolic equation with large lower-order terms.
{"title":"Almost-periodic ground state of the non-self-adjoint Jacobi operator and its applications","authors":"Xing Liang , Hongze Wang , Qi Zhou","doi":"10.1016/j.matpur.2025.103845","DOIUrl":"10.1016/j.matpur.2025.103845","url":null,"abstract":"<div><div>We study the one-dimensional non-self-adjoint Jacobi operators in the almost-periodic media which are “far from” self-adjoint ones. By relaxing the arithmetic condition of the frequency, and regularity conditions of the coefficients, we show the ground states of the operator persists. This result can be seen as complementary of Kozlov's classical result. Besides that, we give two applications: the first is to show the existence and uniqueness of the positive steady state of the discrete Fisher-KPP type equation; the second is to investigate the asymptotic behavior of the discrete stationary parabolic equation with large lower-order terms.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"207 ","pages":"Article 103845"},"PeriodicalIF":2.3,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145941057","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 : 2026-01-02DOI: 10.1016/j.matpur.2025.103844
Dominic Breit , Andrea Cianchi , Daniel Spector
Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type inequalities. As a consequence, they are shown to hold exactly for the same norms as their counterparts depending on the standard gradient operator of the same order. The results offered provide a unified framework for the theory of Sobolev embeddings for the elliptic canceling operators. They build upon and incorporate earlier fundamental contributions dealing with the endpoint case of -norms. They also include previously available results for the symmetric gradient, a prominent instance of an elliptic canceling operator. In particular, the optimal rearrangement-invariant target norm associated with any given domain norm in a Sobolev inequality for any elliptic canceling operator is exhibited. Its explicit form is detected for specific families of rearrangement-invariant spaces, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring are singled out.
{"title":"Sobolev inequalities for canceling operators","authors":"Dominic Breit , Andrea Cianchi , Daniel Spector","doi":"10.1016/j.matpur.2025.103844","DOIUrl":"10.1016/j.matpur.2025.103844","url":null,"abstract":"<div><div>Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type inequalities. As a consequence, they are shown to hold exactly for the same norms as their counterparts depending on the standard gradient operator of the same order. The results offered provide a unified framework for the theory of Sobolev embeddings for the elliptic canceling operators. They build upon and incorporate earlier fundamental contributions dealing with the endpoint case of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norms. They also include previously available results for the symmetric gradient, a prominent instance of an elliptic canceling operator. In particular, the optimal rearrangement-invariant target norm associated with any given domain norm in a Sobolev inequality for any elliptic canceling operator is exhibited. Its explicit form is detected for specific families of rearrangement-invariant spaces, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> are singled out.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"207 ","pages":"Article 103844"},"PeriodicalIF":2.3,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145941056","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 : 2025-12-31DOI: 10.1016/j.matpur.2025.103842
Maciej Borodzik , Paula Truöl
We show that for every genus , there exist quasipositive knots and such that there is a cobordism of genus between and , but there is no ribbon cobordism of genus g in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016.
{"title":"Non-complex cobordisms between quasipositive knots","authors":"Maciej Borodzik , Paula Truöl","doi":"10.1016/j.matpur.2025.103842","DOIUrl":"10.1016/j.matpur.2025.103842","url":null,"abstract":"<div><div>We show that for every genus <span><math><mi>g</mi><mo>≥</mo><mn>0</mn></math></span>, there exist quasipositive knots <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math></span> such that there is a cobordism of genus <span><math><mi>g</mi><mo>=</mo><mo>|</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo>)</mo><mo>−</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>g</mi></mrow></msubsup><mo>)</mo><mo>|</mo></math></span> between <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>g</mi></mrow></msubsup></math></span>, but there is no ribbon cobordism of genus <em>g</em> in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"207 ","pages":"Article 103842"},"PeriodicalIF":2.3,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145941055","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 : 2025-12-31DOI: 10.1016/j.matpur.2025.103841
Wojciech Kucharz
Let X be a compact nonsingular real algebraic set and let M be a compact submanifold of X, with and . Under the assumption , we prove that M can be approximated by nonsingular algebraic subsets of X if and only if certain mod 2 homology classes of X associated with the inclusion map are algebraic. This allows us to give a rather precise description of the approximation properties of k-regulous maps from X to the unit p-sphere in the space of all maps, where k is a nonnegative integer and . A map is called k-regulous if it is of class and its restriction to a Zariski open dense subset of X is a regular map.
{"title":"Algebraic approximation of submanifolds and approximation properties of regulous maps","authors":"Wojciech Kucharz","doi":"10.1016/j.matpur.2025.103841","DOIUrl":"10.1016/j.matpur.2025.103841","url":null,"abstract":"<div><div>Let <em>X</em> be a compact nonsingular real algebraic set and let <em>M</em> be a compact <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> submanifold of <em>X</em>, with <span><math><mtext>dim</mtext><mi>X</mi><mo>=</mo><mi>n</mi></math></span> and <span><math><mtext>dim</mtext><mi>M</mi><mo>=</mo><mi>d</mi></math></span>. Under the assumption <span><math><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>n</mi></math></span>, we prove that <em>M</em> can be approximated by nonsingular algebraic subsets of <em>X</em> if and only if certain mod 2 homology classes of <em>X</em> associated with the inclusion map <span><math><mi>M</mi><mo>↪</mo><mi>X</mi></math></span> are algebraic. This allows us to give a rather precise description of the approximation properties of <em>k</em>-regulous maps from <em>X</em> to the unit <em>p</em>-sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> in the space of all <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> maps, where <em>k</em> is a nonnegative integer and <span><math><mn>2</mn><mi>p</mi><mo>≥</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. A map <span><math><mi>φ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> is called <em>k</em>-regulous if it is of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> and its restriction to a Zariski open dense subset of <em>X</em> is a regular map.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"207 ","pages":"Article 103841"},"PeriodicalIF":2.3,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145897843","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 : 2025-12-15DOI: 10.1016/j.matpur.2025.103832
Lu Chen , Guozhen Lu , Hanli Tang , Bohan Wang
In this paper, we investigate the optimal asymptotic lower bound for the stability of the Sobolev inequality on the Heisenberg group. We first establish the optimal local stability of the Sobolev inequality on the CR sphere by means of bispherical harmonics and a refined orthogonality technique (see Lemma 3.1). The absence of both the Pólya–Szegö inequality and the Riesz rearrangement inequality on the Heisenberg group makes it impossible to apply any rearrangement flow method—either differential or integral—to deduce the global optimal stability of the Sobolev inequality on the CR sphere from its corresponding local stability. To overcome this difficulty, we develop a new approach based on the CR Yamabe flow, which enables us to pass from local to global stability and thereby establish the optimal stability of the Sobolev inequality on the Heisenberg group, with dimension-dependent constants (see Theorem 1.1). As an application, we also obtain the optimal stability of the Hardy–Littlewood–Sobolev (HLS) inequality for a special conformal index, again with dimension-dependent constants (see Theorem 1.2). Our approach is free of any rearrangement argument and can be applied to study the optimal stability problem for the fractional Sobolev or HLS inequalities on the Heisenberg group, once the corresponding continuous flow is established.
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