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Ergodic mean field games: existence of local minimizers up to the Sobolev critical case 遍历均值场博弈：索博列夫临界情况下局部最小值的存在性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-13 DOI: 10.1007/s00526-024-02744-2
Marco Cirant, Alessandro Cosenza, Gianmaria Verzini
{"title":"Ergodic mean field games: existence of local minimizers up to the Sobolev critical case","authors":"Marco Cirant, Alessandro Cosenza, Gianmaria Verzini","doi":"10.1007/s00526-024-02744-2","DOIUrl":"https://doi.org/10.1007/s00526-024-02744-2","url":null,"abstract":"<p>We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side 带右手边的两相 p(x)-Laplacian 问题的平面自由边界的正则性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-13 DOI: 10.1007/s00526-024-02741-5
Fausto Ferrari, Claudia Lederman
{"title":"Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side","authors":"Fausto Ferrari, Claudia Lederman","doi":"10.1007/s00526-024-02741-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02741-5","url":null,"abstract":"<p>We consider viscosity solutions to two-phase free boundary problems for the <i>p</i>(<i>x</i>)-Laplacian with non-zero right hand side. We prove that flat free boundaries are <span>(C^{1,gamma })</span>. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the <i>p</i>(<i>x</i>)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when <span>(p(x)equiv p)</span>, i.e., for the <i>p</i>-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"24 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in $$mathbb {R}^2$$ 带规范约束的变分问题与 $$mathbb {R}^2$ 中半线性椭圆方程的基态之间的关系
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-11 DOI: 10.1007/s00526-024-02710-y
Masato Hashizume
{"title":"Relationship between variational problems with norm constraints and ground state of semilinear elliptic equations in $$mathbb {R}^2$$","authors":"Masato Hashizume","doi":"10.1007/s00526-024-02710-y","DOIUrl":"https://doi.org/10.1007/s00526-024-02710-y","url":null,"abstract":"<p>In this paper, we investigate variational problems in <span>(mathbb {R}^2)</span> with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"341 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On nonminimizing solutions of elliptic free boundary problems 论椭圆自由边界问题的非最小化解
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-07 DOI: 10.1007/s00526-024-02739-z
Kanishka Perera
{"title":"On nonminimizing solutions of elliptic free boundary problems","authors":"Kanishka Perera","doi":"10.1007/s00526-024-02739-z","DOIUrl":"https://doi.org/10.1007/s00526-024-02739-z","url":null,"abstract":"<p>We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and subcritical superlinear free boundary problems, and establish full regularity of the free boundary in dimension <span>(N = 2)</span> and partial regularity in higher dimensions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140886114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems 一类积分微分椭圆型问题的存在性、符号和渐近行为
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02730-8
Márcio A. L. Bahia, Marcos T. O. Pimenta, João R. Santos Junior
{"title":"Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems","authors":"Márcio A. L. Bahia, Marcos T. O. Pimenta, João R. Santos Junior","doi":"10.1007/s00526-024-02730-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02730-8","url":null,"abstract":"<p>In this work we study existence, sign and asymptotic behaviour of solutions for a class of elliptic problems of the integral-differential type under the presence of a parameter. A careful analysis of the influence of the referred parameter on the structure of the set of solutions is made, by considering different reaction terms. Among our main contributions are: (1) a positive answer to Remark 2.4 in Allegretto and Barabanova (Proc R Soc Edinb A 126(3):643–663, 1996); (2) a detailed treatment of the associated eigenvalue problem; (3) The first result involving the existence of a ground-state solution for this class of problems.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884408","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field 向列液晶胶体在外加磁场作用下的线能和面能趋同
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02717-5
François Alouges, Antonin Chambolle, Dominik Stantejsky
{"title":"Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field","authors":"François Alouges, Antonin Chambolle, Dominik Stantejsky","doi":"10.1007/s00526-024-02717-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02717-5","url":null,"abstract":"<p>We use the Landau-de Gennes energy to describe a particle immersed into nematic liquid crystals with a constant applied magnetic field. We derive a limit energy in a regime where both line and point defects are present, showing quantitatively that the close-to-minimal energy is asymptotically concentrated on lines and surfaces nearby or on the particle. We also discuss regularity of minimizers and optimality conditions for the limit energy.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces 部分超定问题和毛细CMC超曲面的刚性和定量稳定性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02733-5
Xiaohan Jia, Zheng Lu, Chao Xia, Xuwen Zhang
{"title":"Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces","authors":"Xiaohan Jia, Zheng Lu, Chao Xia, Xuwen Zhang","doi":"10.1007/s00526-024-02733-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02733-5","url":null,"abstract":"<p>In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the second part, we prove quantitative stability results for the Serrin-type partially overdetermined problem, as well as capillary almost constant mean curvature hypersurfaces in the half-space.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"19 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An exceptional property of the one-dimensional Bianchi–Egnell inequality 一维比安奇-埃格奈尔不等式的一个特殊性质
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02732-6
Tobias König
{"title":"An exceptional property of the one-dimensional Bianchi–Egnell inequality","authors":"Tobias König","doi":"10.1007/s00526-024-02732-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02732-6","url":null,"abstract":"<p>In this paper, for <span>(d ge 1)</span> and <span>(s in (0,frac{d}{2}))</span>, we study the Bianchi–Egnell quotient </p><span>$$begin{aligned} {mathcal {Q}}(f) = inf _{f in dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}} frac{Vert (-Delta )^{s/2} fVert _{L^2({mathbb {R}}^d)}^2 - S_{d,s} Vert fVert _{L^{frac{2d}{d-2s}}(mathbb R^d)}^2}{text {dist}_{dot{H}^s({mathbb {R}}^d)}(f, {mathcal {B}})^2}, qquad f in dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}, end{aligned}$$</span><p>where <span>(S_{d,s})</span> is the best Sobolev constant and <span>({mathcal {B}})</span> is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when <span>(d = 1)</span>, there is a neighborhood of <span>({mathcal {B}})</span> on which the quotient <span>({mathcal {Q}}(f))</span> is larger than the lowest value attainable by sequences converging to <span>({mathcal {B}})</span>. This behavior is surprising because it is contrary to the situation in dimension <span>(d ge 2)</span> described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for <span>(d = 1)</span>, <span>({mathcal {Q}}(f))</span> has no minimizer on <span>(dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}})</span>, which again would be contrary to the situation in <span>(d ge 2)</span>. As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every <span>(d ge 1)</span>. For <span>(d ge 2)</span>, this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For <span>(d =1)</span> we make some numerical observations which support the conjecture stated above.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a novel gradient flow structure for the aggregation equation 关于聚集方程的新型梯度流结构
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02692-x
A. Esposito, R. S. Gvalani, A. Schlichting, M. Schmidtchen
{"title":"On a novel gradient flow structure for the aggregation equation","authors":"A. Esposito, R. S. Gvalani, A. Schlichting, M. Schmidtchen","doi":"10.1007/s00526-024-02692-x","DOIUrl":"https://doi.org/10.1007/s00526-024-02692-x","url":null,"abstract":"<p>The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous inelastic Boltzmann equation, a formal Taylor expansion reveals a link between this equation and the aggregation equation with an appropriately chosen interaction potential. Inspired by this formal link and the fact that the associated aggregation equation also dissipates the kinetic energy, we present a novel way of interpreting the aggregation equation as a gradient flow, in the sense of curves of maximal slope, of the kinetic energy, rather than the usual interaction energy, with respect to an appropriately constructed transportation metric on the space of probability measures.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"107 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A class of fourth-order dispersive wave equations with exponential source 一类指数源四阶色散波方程
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02731-7
Tran Quang Minh, Hong-Danh Pham, Mirelson M. Freitas
{"title":"A class of fourth-order dispersive wave equations with exponential source","authors":"Tran Quang Minh, Hong-Danh Pham, Mirelson M. Freitas","doi":"10.1007/s00526-024-02731-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02731-7","url":null,"abstract":"<p>This paper is concerned with a class of fourth-order dispersive wave equations with <i>exponential</i> source term. Firstly, by applying the contraction mapping principle, we establish the local existence and uniqueness of the solution. In the spirit of the variational principle and mountain pass theorem, a natural phase space is precisely divided into three different energy levels. Then we introduce a family of potential wells to derive a threshold of the existence of global solutions and blow up in finite time of solution in both cases with sub-critical and critical initial energy. These results can be used to extend the previous result obtained by Alves and Cavalcanti (Calc. Var. Partial Differ. Equ. 34 (2009) 377–411). Moreover, an explicit sufficient condition for initial data leading to blow up result is established at an arbitrarily positive initial energy level.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"54 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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