Pub Date : 2026-06-01Epub Date: 2026-02-06DOI: 10.1016/j.matpur.2026.103876
Salvatore Floccari , Lie Fu
We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties of O'Grady type arising as crepant resolutions of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety of OG6-type as above, and all of its powers.
{"title":"The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties","authors":"Salvatore Floccari , Lie Fu","doi":"10.1016/j.matpur.2026.103876","DOIUrl":"10.1016/j.matpur.2026.103876","url":null,"abstract":"<div><div>We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> of O'Grady type arising as crepant resolutions <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>→</mo><mi>K</mi></math></span> of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety <span><math><mover><mrow><mi>K</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> of OG6-type as above, and all of its powers.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103876"},"PeriodicalIF":2.3,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146193142","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 : 2026-06-01Epub 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":"2026-06-01","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 : 2026-06-01Epub Date: 2026-02-06DOI: 10.1016/j.matpur.2026.103875
Dominic Breit , Eduard Feireisl , Martina Hofmanová , Piotr B. Mucha
We study the compressible Navier–Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.
{"title":"Compressible fluids excited by space-dependent transport noise","authors":"Dominic Breit , Eduard Feireisl , Martina Hofmanová , Piotr B. Mucha","doi":"10.1016/j.matpur.2026.103875","DOIUrl":"10.1016/j.matpur.2026.103875","url":null,"abstract":"<div><div>We study the compressible Navier–Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103875"},"PeriodicalIF":2.3,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146193141","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 : 2026-06-01Epub 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":"2026-06-01","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 : 2026-06-01Epub Date: 2026-02-09DOI: 10.1016/j.matpur.2026.103878
Geng Lai , Wancheng Sheng
This paper investigates a supersonic reacting jet issuing from a two-dimensional, finite-length divergent nozzle. To model the jet, a free boundary value problem for the steady two-dimensional Zeldovdovich-von Neumann-Döring (ZND) combustion equations is formulated. The flow is assumed to satisfy a slip condition along the nozzle walls, with its state prescribed at the nozzle inlet. Two external conditions are examined. First, when the nozzle exhausts into a vacuum, the global existence of a supersonic reacting jet is established using the method of characteristics. Furthermore, a sufficient condition on the nozzle wall geometry is provided that leads to the formation of a vacuum region inside the nozzle; it is shown that any such vacuum region must be adjacent to one wall. Second, when the nozzle is surrounded by a static atmosphere whose pressure is lower than the jet pressure at the exit, the analysis demonstrates the formation of singularities (i.e., shock waves) in the jet. This result provides a rigorous mathematical confirmation of the physical phenomenon described in Sec. 148 of the classical text Supersonic Flow and Shock Waves.
{"title":"Supersonic reacting jet flows from two-dimensional divergent nozzles","authors":"Geng Lai , Wancheng Sheng","doi":"10.1016/j.matpur.2026.103878","DOIUrl":"10.1016/j.matpur.2026.103878","url":null,"abstract":"<div><div>This paper investigates a supersonic reacting jet issuing from a two-dimensional, finite-length divergent nozzle. To model the jet, a free boundary value problem for the steady two-dimensional Zeldovdovich-von Neumann-Döring (ZND) combustion equations is formulated. The flow is assumed to satisfy a slip condition along the nozzle walls, with its state prescribed at the nozzle inlet. Two external conditions are examined. First, when the nozzle exhausts into a vacuum, the global existence of a supersonic reacting jet is established using the method of characteristics. Furthermore, a sufficient condition on the nozzle wall geometry is provided that leads to the formation of a vacuum region inside the nozzle; it is shown that any such vacuum region must be adjacent to one wall. Second, when the nozzle is surrounded by a static atmosphere whose pressure is lower than the jet pressure at the exit, the analysis demonstrates the formation of singularities (i.e., shock waves) in the jet. This result provides a rigorous mathematical confirmation of the physical phenomenon described in Sec. 148 of the classical text Supersonic Flow and Shock Waves.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103878"},"PeriodicalIF":2.3,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146193143","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 : 2026-05-01Epub Date: 2026-02-06DOI: 10.1016/j.matpur.2026.103879
Daomin Cao , Yuxia Guo , Shaolong Peng
<div><div>The objective of this paper is to investigate the uniqueness theorems for solutions of the following mixed-order semi-linear elliptic system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>:<span><span><span>(0.1)</span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> may change signs, <span><math><msub><mrow><mi>f
{"title":"Uniqueness theorems for solutions of mixed order elliptic system with general nonlinearity on R4","authors":"Daomin Cao , Yuxia Guo , Shaolong Peng","doi":"10.1016/j.matpur.2026.103879","DOIUrl":"10.1016/j.matpur.2026.103879","url":null,"abstract":"<div><div>The objective of this paper is to investigate the uniqueness theorems for solutions of the following mixed-order semi-linear elliptic system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>:<span><span><span>(0.1)</span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> may change signs, <span><math><msub><mrow><mi>f","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"209 ","pages":"Article 103879"},"PeriodicalIF":2.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146175326","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 : 2026-05-01Epub Date: 2026-02-05DOI: 10.1016/j.matpur.2026.103871
Xun Niu , Kaizhi Wang , Yong Li
This paper establishes several weak KAM type results for Hamiltonian systems with infinite dimensional normal directions, which we call infinite co-dimensional Hamiltonian systems. We first study the Mather's minimization problem for a class of infinite co-dimensional Hamiltonians. Then we prove the existence of weak solutions of the corresponding Hamilton-Jacobi equation which satisfy the equation σ-almost everywhere, where σ is a projected minimal measure satisfying a divergence equation. The abovementioned weak solution transforms the infinite co-dimensional Hamiltonian system into a system with an integrable structure in the weak sense. Finally, we apply our results to a class of one-dimensional nonlinear Schrödinger equations.
{"title":"Weak KAM theory for infinite co-dimensional Hamiltonian systems","authors":"Xun Niu , Kaizhi Wang , Yong Li","doi":"10.1016/j.matpur.2026.103871","DOIUrl":"10.1016/j.matpur.2026.103871","url":null,"abstract":"<div><div>This paper establishes several weak KAM type results for Hamiltonian systems with infinite dimensional normal directions, which we call infinite co-dimensional Hamiltonian systems. We first study the Mather's minimization problem for a class of infinite co-dimensional Hamiltonians. Then we prove the existence of weak solutions of the corresponding Hamilton-Jacobi equation which satisfy the equation <em>σ</em>-almost everywhere, where <em>σ</em> is a projected minimal measure satisfying a divergence equation. The abovementioned weak solution transforms the infinite co-dimensional Hamiltonian system into a system with an integrable structure in the weak sense. Finally, we apply our results to a class of one-dimensional nonlinear Schrödinger equations.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"209 ","pages":"Article 103871"},"PeriodicalIF":2.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146175327","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 : 2026-05-01Epub Date: 2026-02-05DOI: 10.1016/j.matpur.2026.103870
Zongming Guo , Jiayu Li , Fangshu Wan
Existence and non-existence for weighted fourth order elliptic problems in exterior domains are studied. The Liouville type results are obtained for the subcritical case and a unique nontrivial nonnegative radial fast-decay solution is obtained for the supercritical case. It is also seen that the nontrivial nonnegative radial solution does not exist for the critical case.
{"title":"Existence and non-existence for weighted fourth order elliptic problems in exterior domains","authors":"Zongming Guo , Jiayu Li , Fangshu Wan","doi":"10.1016/j.matpur.2026.103870","DOIUrl":"10.1016/j.matpur.2026.103870","url":null,"abstract":"<div><div>Existence and non-existence for weighted fourth order elliptic problems in exterior domains are studied. The Liouville type results are obtained for the subcritical case and a unique nontrivial nonnegative radial fast-decay solution is obtained for the supercritical case. It is also seen that the nontrivial nonnegative radial solution does not exist for the critical case.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"209 ","pages":"Article 103870"},"PeriodicalIF":2.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146175325","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 : 2026-05-01Epub 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":"2026-05-01","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 : 2026-05-01Epub Date: 2026-02-06DOI: 10.1016/j.matpur.2026.103865
Manuel Friedrich , Leonard Kreutz , Konstantinos Zemas
We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work [34] on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of [60], by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.
{"title":"Derivation of Kirchhoff-type plate theories for elastic materials with voids","authors":"Manuel Friedrich , Leonard Kreutz , Konstantinos Zemas","doi":"10.1016/j.matpur.2026.103865","DOIUrl":"10.1016/j.matpur.2026.103865","url":null,"abstract":"<div><div>We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work <span><span>[34]</span></span> on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of <span><span>[60]</span></span>, by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"209 ","pages":"Article 103865"},"PeriodicalIF":2.3,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146175328","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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