Calculus of Variations and Partial Differential Equations最新文献

{"title":"Local uniqueness of ground states for the generalized Choquard equation","authors":"Vladimir Georgiev, Mirko Tarulli, G. Venkov","doi":"10.1007/s00526-024-02742-4","DOIUrl":"https://doi.org/10.1007/s00526-024-02742-4","url":null,"abstract":"","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141117255","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}

{"title":"Plateau’s problem via the Allen–Cahn functional","authors":"Marco A. M. Guaraco, Stephen Lynch","doi":"10.1007/s00526-024-02740-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02740-6","url":null,"abstract":"<p>Let <span>(Gamma )</span> be a compact codimension-two submanifold of <span>({mathbb {R}}^n)</span>, and let <i>L</i> be a nontrivial real line bundle over <span>(X = {mathbb {R}}^n {setminus } Gamma )</span>. We study the Allen–Cahn functional, </p><span>$$begin{aligned}E_varepsilon (u) = int _X varepsilon frac{|nabla u|^2}{2} + frac{(1-|u|^2)^2}{4varepsilon },dx, end{aligned}$$</span><p>on the space of sections <i>u</i> of <i>L</i>. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to <span>(Gamma )</span>. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as <span>(varepsilon rightarrow 0)</span>, the associated family of energy measures converges to an integer rectifiable <span>((n-1))</span>-varifold <i>V</i>. Moreover, <i>V</i> is stationary with respect to any variation which leaves <span>(Gamma )</span> fixed. Away from <span>(Gamma )</span>, this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary <span>(Gamma )</span>. Under additional hypotheses, we can say more about <i>V</i>. When <i>V</i> arises as a limit of critical sections with uniformly bounded Morse index, <span>(Sigma := {{,textrm{supp},}}Vert VVert )</span> is a minimal hypersurface, smooth away from <span>(Gamma )</span> and a singular set of Hausdorff dimension at most <span>(n-8)</span>. If the sections are globally energy minimizing and <span>(n = 3)</span>, then <span>(Sigma )</span> is a smooth surface with boundary, <span>(partial Sigma = Gamma )</span> (at least if <i>L</i> is chosen correctly), and <span>(Sigma )</span> has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau’s problem admits a solution for every boundary curve in <span>({mathbb {R}}^3)</span>. This also works if <span>(4 le nle 7)</span> and <span>(Gamma )</span> is assumed to lie in a strictly convex hypersurface.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941280","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"On a theorem by Schlenk","authors":"Yannis Bähni","doi":"10.1007/s00526-024-02738-0","DOIUrl":"https://doi.org/10.1007/s00526-024-02738-0","url":null,"abstract":"<p>In this paper we prove a generalisation of Schlenk’s theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk’s theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. In this case, the lower bound for the displacement energy is explicitly given by the difference of the two periods. This theorem can be applied to many physical systems including the Hénon–Heiles Hamiltonian and Stark–Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on <span>({mathbb {R}}^{2n})</span> and that the Hofer metric is indeed a metric.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884136","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}

{"title":"Existence and decays of solutions for fractional Schrödinger equations with general potentials","authors":"Yinbin Deng, Shuangjie Peng, Xian Yang","doi":"10.1007/s00526-024-02728-2","DOIUrl":"https://doi.org/10.1007/s00526-024-02728-2","url":null,"abstract":"<p>We revisit the following fractional Schrödinger equation </p><span>$$begin{aligned} varepsilon ^{2s}(-Delta )^su +Vu=u^{p-1},,,,u&gt;0, textrm{in} {mathbb {R}}^N, end{aligned}$$</span>(0.1)<p>where <span>(varepsilon &gt;0)</span> is a small parameter, <span>((-Delta )^s)</span> denotes the fractional Laplacian, <span>(sin (0,1))</span>, <span>(pin (2, 2_s^*))</span>, <span>(2_s^*=frac{2N}{N-2s})</span>, <span>(N&gt;2s)</span>, <span>(Vin Cbig ({mathbb {R}}^N, [0, +infty )big ))</span> is a general potential. Under various assumptions on <i>V</i>(<i>x</i>) at infinity, including <i>V</i>(<i>x</i>) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value <span>(p_*)</span>, such that the above problem admits positive concentration solutions if <span>(pin (p_*, ,2_s^*))</span>, while it has no positive weak solutions for <span>(pin (2,,p_*))</span> if <span>(p_*&gt;2)</span>, where the threshold <span>(p_*in [2, 2^*_s))</span> can be characterized explicitly by</p><span>$$begin{aligned} p_*=left{ begin{array}{ll} 2+frac{2s}{N-2s} &amp;{}quad text{ if } lim limits _{|x| rightarrow infty } (1+|x|^{2s})V(x)=0, 2+frac{omega }{N+2s-omega } &amp;{}quad text{ if } 0!&lt;!inf (1!+!|x|^omega )V(x)!le ! sup (1!+!|x|^omega )V(x)!&lt;! infty text{ for } text{ some } omega !in ! [0, 2s], 2&amp;{}quad text{ if } inf V(x)log (e+|x|^2)&gt;0. end{array}right. end{aligned}$$</span><p>Moreover, corresponding to the various decay assumptions of <i>V</i>(<i>x</i>), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885852","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}