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A Legendre–Fenchel Identity for the Nonlinear Schrödinger Equations on $$mathbb {R}^dtimes mathbb {T}^m$$ : Theory and Applications $$mathbb {R}^dtimes mathbb {T}^m$$ 上非线性薛定谔方程的 Legendre-Fenchel Identity : 理论与应用
The Journal of Geometric Analysis Pub Date : 2024-08-13 DOI: 10.1007/s12220-024-01746-y
Yongming Luo
{"title":"A Legendre–Fenchel Identity for the Nonlinear Schrödinger Equations on $$mathbb {R}^dtimes mathbb {T}^m$$ : Theory and Applications","authors":"Yongming Luo","doi":"10.1007/s12220-024-01746-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01746-y","url":null,"abstract":"<p>The present paper is inspired by a previous work of the author, where the large data scattering problem for the focusing cubic nonlinear Schrödinger equation (NLS) on <span>(mathbb {R}^2times mathbb {T})</span> was studied. Nevertheless, the results from the companion paper are by no means sharp, as we could not even prove the existence of ground state solutions on the formulated threshold. By making use of the semivirial-vanishing geometry, we establish in this paper the sharpened scattering results. Yet due to the mass-critical nature of the model, we encounter the major challenge that the standard scaling arguments fail to perturb the energy functionals. We overcome this difficulty by proving a crucial Legendre–Fenchel identity for the variational problems with prescribed mass and frequency. More precisely, we build up a general framework based on the Legendre–Fenchel identity and show that the much harder or even unsolvable variational problem with prescribed mass, can in fact be equivalently solved by considering the much easier variational problem with prescribed frequency. As an application showing how the geometry of the domain affects the existence of the ground state solutions, we also prove that while all mass-critical ground states on <span>(mathbb {R}^d)</span> must possess the fixed mass <span>({widehat{M}}(Q))</span>, the existence of mass-critical ground states on <span>(mathbb {R}^dtimes mathbb {T})</span> is ensured for a sequence of mass numbers approaching zero.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Large Energy Bubble Solutions for Supercritical Fractional Schrödinger Equation with Double Potentials 具有双重势能的超临界分数薛定谔方程的大能量气泡解决方案
The Journal of Geometric Analysis Pub Date : 2024-08-13 DOI: 10.1007/s12220-024-01769-5
Ting Liu
{"title":"Large Energy Bubble Solutions for Supercritical Fractional Schrödinger Equation with Double Potentials","authors":"Ting Liu","doi":"10.1007/s12220-024-01769-5","DOIUrl":"https://doi.org/10.1007/s12220-024-01769-5","url":null,"abstract":"<p>We consider the following supercritical fractional Schrödinger equation: </p><span>$$begin{aligned} {left{ begin{array}{ll} (-Delta )^s u + V(y) u=Q(y)u^{2_s^*-1+varepsilon }, ;u&gt;0, &amp;{}hbox { in } {mathbb {R}}^{N}, u in D^s( {mathbb {R}}^{N}), end{array}right. } end{aligned}$$</span>(*)<p>where <span>(2_s^*=frac{2N}{N-2s},; N&gt; 4s)</span>, <span>(0&lt; s &lt; 1)</span>, <span>((y',y'') in {mathbb {R}}^{2} times {mathbb {R}}^{N-2})</span>, <span>(V(y) = V(|y'|,y''))</span> and <span>(Q(y) = Q(|y'|,y'') not equiv 0)</span> are two bounded non-negative functions. Under some suitable assumptions on the potentials <i>V</i> and <i>Q</i>, we will use the finite-dimensional reduction argument and some local Pohozaev type identities to prove that for <span>(varepsilon &gt; 0)</span> small enough, the problem <span>((*))</span> has a large number of bubble solutions whose functional energy is in the order <span>(varepsilon ^{-frac{N-4s}{(N-2s)^2}}.)</span>\u0000</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spectral Stability of Constrained Solitary Waves for the Generalized Singular Perturbed KdV Equation 广义奇异扰动 KdV 方程受约束孤波的频谱稳定性
The Journal of Geometric Analysis Pub Date : 2024-08-12 DOI: 10.1007/s12220-024-01757-9
Fangyu Han, Yuetian Gao
{"title":"Spectral Stability of Constrained Solitary Waves for the Generalized Singular Perturbed KdV Equation","authors":"Fangyu Han, Yuetian Gao","doi":"10.1007/s12220-024-01757-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01757-9","url":null,"abstract":"<p>This paper is systematically concerned with the solitary waves on the constrained manifold preserved the <span>(L^2)</span>-momentum conservation for the generalized singular perturbed KdV equation with <span>(L^2)</span><i>-subcritical, critical and supercritical nonlinearities</i>, which is a long-wave approximation to the capillary-gravity waves in an infinitely long channel with a flat bottom. First, using the profile decomposition in <span>(H^2)</span> and the optimal Gagliardo–Nirenberg inequality, we prove the existence of subcritical ground state solitary waves and describe their asymptotic behavior. Second, we obtain some sufficient conditions for the existence and non-existence of critical ground states, and then prove the existence of critical and supercritical ground state solitary waves on the Derrick–Pohozaev manifold by utilizing the new minimax argument and the numerical simulation of the best Gagliardo–Nirenberg embedding constant. Meanwhile, we use the moving plane method to obtain the existence of positive and radially symmetric solutions. Furthermore, we study the concentration behavior of the critical ground state solutions. Finally, the spectral stability of the ground state solitary wave solutions is discussed by using the instability index theorem.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"54 63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Solutions to the Generalized Dual Minkowski Problem 广义二元闵科夫斯基问题的解的存在性
The Journal of Geometric Analysis Pub Date : 2024-08-12 DOI: 10.1007/s12220-024-01754-y
Mingyang Li, YanNan Liu, Jian Lu
{"title":"Existence of Solutions to the Generalized Dual Minkowski Problem","authors":"Mingyang Li, YanNan Liu, Jian Lu","doi":"10.1007/s12220-024-01754-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01754-y","url":null,"abstract":"<p>Given a real number <i>q</i> and a star body in the <i>n</i>-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for <span>(q&lt;0)</span>, and the even generalized dual Minkowski problem for <span>(0le qle 1)</span>. We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for <span>(1&lt;q&lt;n)</span>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Minimal Laminations and Level Sets of 1-Harmonic Functions 1 次谐函数的最小层叠和水平集
The Journal of Geometric Analysis Pub Date : 2024-08-08 DOI: 10.1007/s12220-024-01758-8
Aidan Backus
{"title":"Minimal Laminations and Level Sets of 1-Harmonic Functions","authors":"Aidan Backus","doi":"10.1007/s12220-024-01758-8","DOIUrl":"https://doi.org/10.1007/s12220-024-01758-8","url":null,"abstract":"<p>We collect several results concerning regularity of minimal laminations, and governing the various modes of convergence for sequences of minimal laminations. We then apply this theory to prove that a function has locally least gradient (is 1-harmonic) iff its level sets are a minimal lamination; this resolves an open problem of Daskalopoulos and Uhlenbeck.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Cut Locus of Submanifolds of a Finsler Manifold 论芬斯勒流形子流形的切点
The Journal of Geometric Analysis Pub Date : 2024-08-08 DOI: 10.1007/s12220-024-01751-1
Aritra Bhowmick, Sachchidanand Prasad
{"title":"On the Cut Locus of Submanifolds of a Finsler Manifold","authors":"Aritra Bhowmick, Sachchidanand Prasad","doi":"10.1007/s12220-024-01751-1","DOIUrl":"https://doi.org/10.1007/s12220-024-01751-1","url":null,"abstract":"<p>In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and Prasad (Algebr Geom Topol 23(9):4185–4233, 2023). Given a submanifold <i>N</i>, we consider an <i>N</i>-geodesic loop as an <i>N</i>-geodesic starting and ending in <i>N</i>, possibly at different points. This class of geodesics were studied by Omori (J Differ Geom 2:233–252, 1968). We obtain a generalization of Klingenberg’s lemma for closed geodesics (Klingenberg in: Ann Math 2(69):654–666, 1959). for <i>N</i>-geodesic loops in the reversible Finsler setting.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Blow-Up of Solutions for the Fourth-Order Schrödinger Equation with Combined Power-Type Nonlinearities 具有组合功率型非线性的四阶薛定谔方程的炸裂解
The Journal of Geometric Analysis Pub Date : 2024-08-06 DOI: 10.1007/s12220-024-01747-x
Zaiyun Zhang, Dandan Wang, Jiannan Chen, Zihan Xie, Chengzhao Xu
{"title":"Blow-Up of Solutions for the Fourth-Order Schrödinger Equation with Combined Power-Type Nonlinearities","authors":"Zaiyun Zhang, Dandan Wang, Jiannan Chen, Zihan Xie, Chengzhao Xu","doi":"10.1007/s12220-024-01747-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01747-x","url":null,"abstract":"<p>In this paper, we mainly consider the blow-up solutions of the fourth-order Schrödinger equation with combined power-type nonlinearities </p><span>$$begin{aligned} iu_{t}+alpha Delta ^{2}u+beta Delta u+lambda _{1}left| u right| ^{sigma _{1}}u+lambda _{2}left| u right| ^{sigma _{2}}u=0, end{aligned}$$</span><p>where <span>(4&lt;n&lt;8,)</span> <span>(beta =left{ { 0, 1}right} , alpha ,,lambda _{1}in mathbb {R})</span> and <span>(lambda _{2}&lt;0)</span>. Firstly, using Banach’s fixed point theorem, iterative method and nonlinear estimates, we establish the local well-posedness of solutions with the initial data <span>(u_{0}in H^{2}(mathbb {R}^{n}))</span>. Then, based on variational analysis theory for dynamical system, using localized Virial identity, we establish a new Morawetz estimates and upper bound estimates to prove the existence of blow-up solutions in finite time. Finally, applying the local well-posedness above, we demonstrate the blow-up criteria of solutions and prove it by contradiction method.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniform Metric Graphs 统一度量图
The Journal of Geometric Analysis Pub Date : 2024-08-05 DOI: 10.1007/s12220-024-01735-1
David A. Herron
{"title":"Uniform Metric Graphs","authors":"David A. Herron","doi":"10.1007/s12220-024-01735-1","DOIUrl":"https://doi.org/10.1007/s12220-024-01735-1","url":null,"abstract":"<p>We prove that every complete metric space “is” the boundary of a uniform length space whose quasihyperbolization is a geodesic visual Gromov hyperbolic space. There is a natural quasimöbius identification of the original space’s conformal gauge with the canonical gauge on the Gromov boundary. All parameters are absolute constants.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Homogeneous Projectively Flat Finsler Metrics 论同质投影平坦芬斯勒度量
The Journal of Geometric Analysis Pub Date : 2024-08-02 DOI: 10.1007/s12220-024-01752-0
A. Tayebi, B. Najafi
{"title":"On Homogeneous Projectively Flat Finsler Metrics","authors":"A. Tayebi, B. Najafi","doi":"10.1007/s12220-024-01752-0","DOIUrl":"https://doi.org/10.1007/s12220-024-01752-0","url":null,"abstract":"<p>Recently, Liu-Deng studied projectively flat homogeneous <span>((alpha , beta ))</span>-metrics and showed that if these metrics are not Riemannian nor locally Minkowskian, then the Finsler metrics are left invariant Randers metrics on the hyperbolic space <span>(textbf{H}^n)</span> as a solvable Lie group (Liu and Deng in Forum Math 27:3149–3165, 2015). In this paper, we study homogeneous projectively flat (or projective) general Finsler metrics. First, we prove that homogeneous projectively flat Finsler metrics have vanishing <span>({{bar{textbf{E}}}})</span>-curvature if and only if they have almost isotropic S-curvature if and only if they have relatively isotropic L-curvature. In any cases, the Finsler metric reduces to a locally Minkowskian metric or a Riemannian metric of constant sectional curvature. This yields a classification of homogeneous projective Finsler metrics with the above mentioned non-Riemannian curvatures properties. Finally, we show that Liu-Deng’s Randers metrics are Douglas metrics which have not isotropic S-curvature nor relatively isotropic L-curvature.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A New Approach for Hardy Spaces on Euclidean Space 欧几里得空间上的哈代空间新方法
The Journal of Geometric Analysis Pub Date : 2024-08-02 DOI: 10.1007/s12220-024-01749-9
Youhai Huang, Qiquan Fang, Xiangxing Tao, Taotao Zheng
{"title":"A New Approach for Hardy Spaces on Euclidean Space","authors":"Youhai Huang, Qiquan Fang, Xiangxing Tao, Taotao Zheng","doi":"10.1007/s12220-024-01749-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01749-9","url":null,"abstract":"<p>In this paper, we develop a new approach to the classical Hardy spaces on Euclidean space. The new ideas of this paper are (i) introduce new test functions and distributions; (ii) we use the classical Calderón reproducing formula on <span>(L^2({mathbb {R}}^d))</span> only; (iii) the Hardy space is defined by the collections of all new distributions with the classical wavelet-type representation and the norms of Hardy space are defined as the norms of the classical atomic Hardy spaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141887382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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