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}
Pub Date : 2025-12-15DOI: 10.1016/j.matpur.2025.103836
Soobin Cho , Panki Kim
We study symmetric Dirichlet forms on metric measure spaces, which may possess both strongly local and pure-jump parts. We introduce a new formulation of a tail condition for jump measures and weighted functional inequalities. Our framework accommodates Dirichlet forms with singular jump measures and those associated with trace processes of mixed-type stable processes. Using these new weighted functional inequalities, we establish stable, equivalent characterizations of Hölder regularity for caloric and harmonic functions. As an application of our main result, we prove the Hölder continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.
{"title":"Stability of Hölder regularity and weighted functional inequalities","authors":"Soobin Cho , Panki Kim","doi":"10.1016/j.matpur.2025.103836","DOIUrl":"10.1016/j.matpur.2025.103836","url":null,"abstract":"<div><div>We study symmetric Dirichlet forms on metric measure spaces, which may possess both strongly local and pure-jump parts. We introduce a new formulation of a tail condition for jump measures and weighted functional inequalities. Our framework accommodates Dirichlet forms with singular jump measures and those associated with trace processes of mixed-type stable processes. Using these new weighted functional inequalities, we establish stable, equivalent characterizations of Hölder regularity for caloric and harmonic functions. As an application of our main result, we prove the Hölder continuity of caloric functions for a large class of symmetric Markov processes exhibiting boundary blow-up behavior, among other results.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103836"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841746","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.2025.103837
Zhentao He, Chao Ji
In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graphs with localized nonlinearities where is the Dirac operator on , , , , is the characteristic function of the compact core , and . First, for , we prove the existence of normalized solutions to (NLDE) using a perturbation argument. Then, for , we present sufficient conditions under which the normalized solutions to (NLDE) exist. Finally, we extend these results to the case and, for all , prove the existence of normalized solutions to (NLDE) when is an eigenvalue of the operator . In the Appendix, we study the influence of the parameters on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.
{"title":"Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities","authors":"Zhentao He, Chao Ji","doi":"10.1016/j.matpur.2025.103837","DOIUrl":"10.1016/j.matpur.2025.103837","url":null,"abstract":"<div><div>In this paper, we study the following nonlinear Dirac equations (NLDE) on noncompact metric graphs <span><math><mi>G</mi></math></span> with localized nonlinearities<span><span><span><math><mi>D</mi><mi>u</mi><mo>−</mo><mi>ω</mi><mi>u</mi><mo>=</mo><mi>a</mi><msub><mrow><mi>χ</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo></math></span></span></span> where <span><math><mi>D</mi></math></span> is the Dirac operator on <span><math><mi>G</mi></math></span>, <span><math><mi>u</mi><mo>:</mo><mi>G</mi><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, <span><math><mi>ω</mi><mo>∈</mo><mi>R</mi></math></span>, <span><math><mi>a</mi><mo>></mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>K</mi></mrow></msub></math></span> is the characteristic function of the compact core <span><math><mi>K</mi></math></span>, and <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>. First, for <span><math><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><mn>4</mn></math></span>, we prove the existence of normalized solutions to (NLDE) using a perturbation argument. Then, for <span><math><mi>p</mi><mo>≥</mo><mn>4</mn></math></span>, we present sufficient conditions under which the normalized solutions to (NLDE) exist. Finally, we extend these results to the case <span><math><mi>a</mi><mo><</mo><mn>0</mn></math></span> and, for all <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span>, prove the existence of normalized solutions to (NLDE) when <span><math><mi>λ</mi><mo>=</mo><mo>−</mo><mi>m</mi><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is an eigenvalue of the operator <span><math><mi>D</mi></math></span>. In the Appendix, we study the influence of the parameters <span><math><mi>m</mi><mo>,</mo><mi>c</mi><mo>></mo><mn>0</mn></math></span> on the existence of normalized solutions to (NLDE). To the best of our knowledge, this is the first study to investigate the normalized solutions to (NLDE) on metric graphs.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103837"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841747","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.2025.103834
Daisuke Naimen
We establish a series of concentration and oscillation estimates for semilinear elliptic equations with exponential nonlinearity in a disc. Especially, we show various new results on the supercritical case which are left open in the previous works. We begin with the concentration analysis of blow-up solutions by extending the scaling and pointwise techniques developed in the previous studies. A striking result is that we detect an infinite sequence of bubbles. The precise characterization of the limit profile, energy, and location of each bubble is given. Moreover, we arrive at a natural interpretation, the infinite sequence of bubbles causes the infinite oscillation of the solutions. Based on this idea and our concentration estimates, we next carry out the oscillation analysis. The results allow us to prove that the intersection number between blow-up solutions and singular functions diverges to infinity. Applying this, we finally demonstrate infinite oscillations of bifurcation diagrams of supercritical equations. We present the results mentioned above through a series of two papers. The present one is devoted to the former part, that is, the concentration analysis.
{"title":"Concentration and oscillation analysis of positive solutions to semilinear elliptic equations with exponential growth in a disc. I","authors":"Daisuke Naimen","doi":"10.1016/j.matpur.2025.103834","DOIUrl":"10.1016/j.matpur.2025.103834","url":null,"abstract":"<div><div>We establish a series of concentration and oscillation estimates for semilinear elliptic equations with exponential nonlinearity <span><math><msup><mrow><mi>e</mi></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></msup></math></span> in a disc. Especially, we show various new results on the supercritical case <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span> which are left open in the previous works. We begin with the concentration analysis of blow-up solutions by extending the scaling and pointwise techniques developed in the previous studies. A striking result is that we detect an infinite sequence of bubbles. The precise characterization of the limit profile, energy, and location of each bubble is given. Moreover, we arrive at a natural interpretation, the infinite sequence of bubbles causes the infinite oscillation of the solutions. Based on this idea and our concentration estimates, we next carry out the oscillation analysis. The results allow us to prove that the intersection number between blow-up solutions and singular functions diverges to infinity. Applying this, we finally demonstrate infinite oscillations of bifurcation diagrams of supercritical equations. We present the results mentioned above through a series of two papers. The present one is devoted to the former part, that is, the concentration analysis.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103834"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792180","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.2025.103833
Paul Gassiat , Florin Suciu
We consider the (sub-Riemannian type) control problem of finding a path going from an initial point x to a target point y, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the bracket-generating (Hörmander) condition, so that the classical Chow-Rashevskii theorem guarantees the existence of such a path. One natural way to try to solve this problem is via a gradient flow on control space. However, since the corresponding dynamics may have saddle points, any convergence result must rely on suitable (e.g. random) initialisation. We consider the case when this initialisation is irregular, which is conveniently formulated via Lyons' rough path theory. We show that one advantage of this initialisation is that the saddle points are moved to infinity, while minima remain at a finite distance from the starting point. In the step 2-nilpotent case, we further manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The proof is based on combining ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our study comes from the training of deep Residual Neural Nets, in the regime when the number of trainable parameters per layer is smaller than the dimension of the data vector.
{"title":"A gradient flow on control space with rough initial condition","authors":"Paul Gassiat , Florin Suciu","doi":"10.1016/j.matpur.2025.103833","DOIUrl":"10.1016/j.matpur.2025.103833","url":null,"abstract":"<div><div>We consider the (sub-Riemannian type) control problem of finding a path going from an initial point <em>x</em> to a target point <em>y</em>, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the bracket-generating (Hörmander) condition, so that the classical Chow-Rashevskii theorem guarantees the existence of such a path. One natural way to try to solve this problem is via a gradient flow on control space. However, since the corresponding dynamics may have saddle points, any convergence result must rely on suitable (e.g. random) initialisation. We consider the case when this initialisation is irregular, which is conveniently formulated via Lyons' rough path theory. We show that one advantage of this initialisation is that the saddle points are moved to infinity, while minima remain at a finite distance from the starting point. In the step 2-nilpotent case, we further manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The proof is based on combining ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our study comes from the training of deep Residual Neural Nets, in the regime when the number of trainable parameters per layer is smaller than the dimension of the data vector.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103833"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760742","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.2025.103838
José Antonio Carrillo , Xuanrui Feng , Shuchen Guo , Pierre-Emmanuel Jabin , Zhenfu Wang
We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.
{"title":"Relative entropy method for particle approximation of the Landau equation for Maxwellian molecules","authors":"José Antonio Carrillo , Xuanrui Feng , Shuchen Guo , Pierre-Emmanuel Jabin , Zhenfu Wang","doi":"10.1016/j.matpur.2025.103838","DOIUrl":"10.1016/j.matpur.2025.103838","url":null,"abstract":"<div><div>We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103838"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145841748","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}
Stochastic incompleteness of a Riemannian manifold M amounts to the nonconservation of probability for the heat semigroup on M. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to for one, hence all, general nonlinearity ψ which is only required to be continuous, nondecreasing, with and in . Similar statements hold for (sub)solutions that may change sign. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation with bounded initial data for one, hence all, general nonlinearity ϕ which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.
{"title":"A general nonlinear characterization of stochastic incompleteness","authors":"Gabriele Grillo , Kazuhiro Ishige , Matteo Muratori , Fabio Punzo","doi":"10.1016/j.matpur.2025.103839","DOIUrl":"10.1016/j.matpur.2025.103839","url":null,"abstract":"<div><div>Stochastic incompleteness of a Riemannian manifold <em>M</em> amounts to the nonconservation of probability for the heat semigroup on <em>M</em>. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to <span><math><mi>Δ</mi><mi>W</mi><mo>=</mo><mi>ψ</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span> for one, hence all, general nonlinearity <em>ψ</em> which is only required to be continuous, nondecreasing, with <span><math><mi>ψ</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mi>ψ</mi><mo>></mo><mn>0</mn></math></span> in <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo></math></span>. Similar statements hold for (sub)solutions that may change sign. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><mi>ϕ</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> with bounded initial data for one, hence all, general nonlinearity <em>ϕ</em> which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"206 ","pages":"Article 103839"},"PeriodicalIF":2.3,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792178","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 : 2025-10-17DOI: 10.1016/j.matpur.2025.103812
Yangmin Xiong , Anna Kostianko , Chunyou Sun , Sergey Zelik
We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damped wave equation with Dirichlet boundary conditions, quintic 3D wave equation with periodic boundary conditions and 2D Navier-Stokes system in a bounded domain. The examples of finite-dimensional uniform attractors for some special external forces which are not translation-compact are also given.
{"title":"Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces","authors":"Yangmin Xiong , Anna Kostianko , Chunyou Sun , Sergey Zelik","doi":"10.1016/j.matpur.2025.103812","DOIUrl":"10.1016/j.matpur.2025.103812","url":null,"abstract":"<div><div>We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damped wave equation with Dirichlet boundary conditions, quintic 3D wave equation with periodic boundary conditions and 2D Navier-Stokes system in a bounded domain. The examples of finite-dimensional uniform attractors for some special external forces which are not translation-compact are also given.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103812"},"PeriodicalIF":2.3,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362036","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 : 2025-10-17DOI: 10.1016/j.matpur.2025.103813
Haoyang Chen , Yi Zhou
This paper is devoted to the study of the global existence of smooth solutions to the 3+1 dimensional Einstein-Klein-Gordon systems with a isometry group for a class of regular large Cauchy data. In our first paper [14], we reduce these Einstein-Klein-Gordon equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof of the global regularity for this 2+1 dimensional system. The major difficulties arise from the lack of smallness at initial time and near centre . To overcome these obstacles, we firstly show the non-concentration of the energy near the first possible singularity, which transforms the large-data problem to a small energy problem. Then, near the first possible singularity, we derive spacetime energy estimates to prove global well-posedness for initial data with small energy.
{"title":"Global regularity for Einstein-Klein-Gordon system with U(1)×R isometry group, II","authors":"Haoyang Chen , Yi Zhou","doi":"10.1016/j.matpur.2025.103813","DOIUrl":"10.1016/j.matpur.2025.103813","url":null,"abstract":"<div><div>This paper is devoted to the study of the global existence of smooth solutions to the 3+1 dimensional Einstein-Klein-Gordon systems with a <span><math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>R</mi></math></span> isometry group for a class of regular large Cauchy data. In our first paper <span><span>[14]</span></span>, we reduce these Einstein-Klein-Gordon equations to a 2+1 dimensional Einstein-wave-Klein-Gordon system. And we show that the first possible singularity can only occur at the axis. In this paper, we give a proof of the global regularity for this 2+1 dimensional system. The major difficulties arise from the lack of smallness at initial time <span><math><mi>t</mi><mo>=</mo><mn>0</mn></math></span> and near centre <span><math><mi>r</mi><mo>=</mo><mn>0</mn></math></span>. To overcome these obstacles, we firstly show the non-concentration of the energy near the first possible singularity, which transforms the large-data problem to a small energy problem. Then, near the first possible singularity, we derive spacetime energy estimates to prove global well-posedness for initial data with small energy.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103813"},"PeriodicalIF":2.3,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145361414","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}
We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain , , for the weight varying in a suitable class of sign-changing bounded functions. Denoting with u the optimal eigenfunction and with D its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of D tends to zero, the unique maximum point of u, , tends to a point of maximal mean curvature of ∂Ω. Furthermore, we show that D is the intersection with Ω of a nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of D.
These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.
{"title":"Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem","authors":"Lorenzo Ferreri , Dario Mazzoleni , Benedetta Pellacci , Gianmaria Verzini","doi":"10.1016/j.matpur.2025.103815","DOIUrl":"10.1016/j.matpur.2025.103815","url":null,"abstract":"<div><div>We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, <span><math><mi>N</mi><mo>≥</mo><mn>2</mn></math></span>, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with <em>u</em> the optimal eigenfunction and with <em>D</em> its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of <em>D</em> tends to zero, the unique maximum point of <em>u</em>, <span><math><mi>P</mi><mo>∈</mo><mo>∂</mo><mi>Ω</mi></math></span>, tends to a point of maximal mean curvature of ∂Ω. Furthermore, we show that <em>D</em> is the intersection with Ω of a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></math></span> nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of <em>D</em>.</div><div>These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103815"},"PeriodicalIF":2.3,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362039","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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