Pub Date : 2026-02-05DOI: 10.1016/j.matpur.2026.103863
William M. Feldman, Zhonggan Huang
We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) regularity in dimension and we show the same regularity result in conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits.
{"title":"Regularity theory of a gradient degenerate Neumann problem","authors":"William M. Feldman, Zhonggan Huang","doi":"10.1016/j.matpur.2026.103863","DOIUrl":"10.1016/j.matpur.2026.103863","url":null,"abstract":"<div><div>We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span> regularity in dimension <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> and we show the same regularity result in <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span> conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"209 ","pages":"Article 103863"},"PeriodicalIF":2.3,"publicationDate":"2026-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146175329","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.
{"title":"Asymptotically sharp stability of Sobolev inequalities on the Heisenberg group with dimension-dependent constants","authors":"Lu Chen , Guozhen Lu , Hanli Tang , Bohan Wang","doi":"10.1016/j.matpur.2025.103832","DOIUrl":"10.1016/j.matpur.2025.103832","url":null,"abstract":"<div><div>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 <span><span>Lemma 3.1</span></span>). 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 <span><span>Theorem 1.1</span></span>). 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 <span><span>Theorem 1.2</span></span>). 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.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103832"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792179","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.2025.103835
Mingxin Wang , Lei Zhang
It is well-known that the principal eigenvalue of the linearized system plays a critical role in investigating the dynamical properties of nonlinear evolution systems with nonlocal dispersal. However, due to the lack of compactness, additional conditions must be imposed on the coefficients to establish the existence of the principal eigenvalue. In this paper, we approximate the generalized principal eigenvalue of a cooperative and irreducible nonlocal dispersal system one that admits a Collatz-Wielandt characterization by constructing monotonic upper and lower control systems with well-defined principal eigenvalues. Furthermore, we demonstrate that the generalized principal eigenvalue functions identically to the conventional principal eigenvalue in terms of its role.
{"title":"Approximation of the generalized principal eigenvalue of cooperative nonlocal dispersal systems and applications","authors":"Mingxin Wang , Lei Zhang","doi":"10.1016/j.matpur.2025.103835","DOIUrl":"10.1016/j.matpur.2025.103835","url":null,"abstract":"<div><div>It is well-known that the principal eigenvalue of the linearized system plays a critical role in investigating the dynamical properties of nonlinear evolution systems with nonlocal dispersal. However, due to the lack of compactness, additional conditions must be imposed on the coefficients to establish the existence of the principal eigenvalue. In this paper, we approximate the generalized principal eigenvalue of a cooperative and irreducible nonlocal dispersal system one that admits a Collatz-Wielandt characterization by constructing monotonic upper and lower control systems with well-defined principal eigenvalues. Furthermore, we demonstrate that the generalized principal eigenvalue functions identically to the conventional principal eigenvalue in terms of its role.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103835"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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