Calculus of Variations and Partial Differential Equations最新文献_第10页

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

引用次数: 0

On a theorem by Schlenk 关于施伦克的一个定理

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02738-0

Yannis Bähni

{"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":"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 ({mathbb {R}}^{2n}) and that the Hofer metric is indeed a metric.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"17 1","pages":""},"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}

引用次数: 0

Existence and decays of solutions for fractional Schrödinger equations with general potentials 具有一般势能的分数薛定谔方程的解的存在性和衰减性

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-05 DOI: 10.1007/s00526-024-02728-2

Yinbin Deng, Shuangjie Peng, Xian Yang

{"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":"We revisit the following fractional Schrödinger equation $$begin{aligned} varepsilon ^{2s}(-Delta )^su +Vu=u^{p-1},,,,u>0, textrm{in} {mathbb {R}}^N, end{aligned}$$(0.1)where (varepsilon >0) is a small parameter, ((-Delta )^s) denotes the fractional Laplacian, (sin (0,1)), (pin (2, 2_s^*)), (2_s^*=frac{2N}{N-2s}), (N>2s), (Vin Cbig ({mathbb {R}}^N, [0, +infty )big )) is a general potential. Under various assumptions on V(x) at infinity, including V(x) 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 (p_*), such that the above problem admits positive concentration solutions if (pin (p_*, ,2_s^*)), while it has no positive weak solutions for (pin (2,,p_*)) if (p_*>2), where the threshold (p_*in [2, 2^*_s)) can be characterized explicitly by$$begin{aligned} p_*=left{ begin{array}{ll} 2+frac{2s}{N-2s} &{}quad text{ if } lim limits _{|x| rightarrow infty } (1+|x|^{2s})V(x)=0, 2+frac{omega }{N+2s-omega } &{}quad text{ if } 0!<!inf (1!+!|x|^omega )V(x)!le ! sup (1!+!|x|^omega )V(x)!<! infty text{ for } text{ some } omega !in ! [0, 2s], 2&{}quad text{ if } inf V(x)log (e+|x|^2)>0. end{array}right. end{aligned}$$Moreover, corresponding to the various decay assumptions of V(x), 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.","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"78 1","pages":""},"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}

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

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":"In this paper, for (d ge 1) and (s in (0,frac{d}{2})), we study the Bianchi–Egnell quotient $$begin{aligned} {mathcal {Q}}(f) = inf _{f in dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}} frac{Vert (-Delta )^{s/2} fVert _{L^2({mathbb {R}}^d)}^2 - S_{d,s} Vert fVert _{L^{frac{2d}{d-2s}}(mathbb R^d)}^2}{text {dist}_{dot{H}^s({mathbb {R}}^d)}(f, {mathcal {B}})^2}, qquad f in dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}, end{aligned}$$where (S_{d,s}) is the best Sobolev constant and ({mathcal {B}}) is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when (d = 1), there is a neighborhood of ({mathcal {B}}) on which the quotient ({mathcal {Q}}(f)) is larger than the lowest value attainable by sequences converging to ({mathcal {B}}). This behavior is surprising because it is contrary to the situation in dimension (d ge 2) described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for (d = 1), ({mathcal {Q}}(f)) has no minimizer on (dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}), which again would be contrary to the situation in (d ge 2). As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every (d ge 1). For (d ge 2), this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For (d =1) we make some numerical observations which support the conjecture stated above.","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

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

引用次数: 0

Traveling waves and effective mass for the regularized Landau-Pekar equations 正则化兰道-佩卡方程的移动波和有效质量

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI: 10.1007/s00526-024-02735-3

Simone Rademacher

引用次数: 0

Global dynamics of large solution for the compressible Navier–Stokes–Korteweg equations 可压缩纳维-斯托克斯-科特韦格方程大解的全局动力学

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI: 10.1007/s00526-024-02723-7

Zihao Song

引用次数: 0

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation 一类具有临界增长和对数扰动的非线性退化椭圆算子的 Dirichlet 问题

IF 2.1 2区数学

Calculus of Variations and Partial Differential Equations Pub Date : 2024-04-26 DOI: 10.1007/s00526-024-02708-6

Hua Chen, Xin Liao, Ming Zhang

引用次数: 0