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Rado’s Theorem for CR Functions on Hypersurfaces 超曲面上 CR 函数的拉多定理
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01763-x
S. Berhanu, Xiaoshan Li
{"title":"Rado’s Theorem for CR Functions on Hypersurfaces","authors":"S. Berhanu, Xiaoshan Li","doi":"10.1007/s12220-024-01763-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01763-x","url":null,"abstract":"<p>We prove a generalization of a well-known theorem of Rado for continuous CR functions on a class of bihololomorphically invariant hypersurfaces that are considerably larger than convex ones of finite type and strictly pseudoconvex hypersurfaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190142","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
Normalized Solutions to N-Laplacian Equations in $${mathbb {R}}^N$$ with Exponential Critical Growth 具有指数临界增长的 $${mathbb {R}}^N$ 中 N 拉普拉斯方程的归一化解
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01771-x
Jingbo Dou, Ling Huang, Xuexiu Zhong
{"title":"Normalized Solutions to N-Laplacian Equations in $${mathbb {R}}^N$$ with Exponential Critical Growth","authors":"Jingbo Dou, Ling Huang, Xuexiu Zhong","doi":"10.1007/s12220-024-01771-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01771-x","url":null,"abstract":"<p>In this paper, we are concerned with normalized solutions <span>((u,lambda )in W^{1,N}(mathbb {R}^N)times mathbb {R}^+)</span> to the following <i>N</i>-Laplacian problem </p><span>$$begin{aligned} -{text {div}}(|nabla u|^{N-2} nabla u)+lambda |u|^{N-2} u=f(u) text{ in } mathbb {R}^N,~N ge 2, end{aligned}$$</span><p>satisfying the normalization constraint <span>(int _{mathbb {R}^N}|u|^Ntextrm{d}x=c^N)</span>. The nonlinearity <i>f</i>(<i>s</i>) is an exponential critical growth function, i.e., behaves like <span>(exp (alpha |s|^{N /(N-1)}))</span> for some <span>(alpha &gt;0)</span> as <span>(|s| rightarrow infty )</span>. Under some mild conditions, we show the existence of normalized mountain pass type solution via the variational method. We also emphasize the normalized ground state solution has a mountain pass characterization under some further assumption. Our existence results in present paper also solve a Soave’s type open problem (J Funct Anal 279(6):108610, 2020) on the nonlinearities having an exponential critical growth.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190145","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 Steklov Eigenvalues Under Volume Constraints 体积约束下的大斯特克洛夫特征值
The Journal of Geometric Analysis Pub Date : 2024-08-19 DOI: 10.1007/s12220-024-01768-6
Alexandre Girouard, Panagiotis Polymerakis
{"title":"Large Steklov Eigenvalues Under Volume Constraints","authors":"Alexandre Girouard, Panagiotis Polymerakis","doi":"10.1007/s12220-024-01768-6","DOIUrl":"https://doi.org/10.1007/s12220-024-01768-6","url":null,"abstract":"<p>In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold with connected boundary diffeomorphic to a product admits a family of Riemannian metrics which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. These are the first examples of Riemannian metrics with these properties on three-dimensional manifolds.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190143","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
Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula 各向异性亚历山德罗夫-芬切尔式不等式和熊-闵科夫斯基公式
The Journal of Geometric Analysis Pub Date : 2024-08-13 DOI: 10.1007/s12220-024-01759-7
Jinyu Gao, Guanghan Li
{"title":"Anisotropic Alexandrov–Fenchel Type Inequalities and Hsiung–Minkowski Formula","authors":"Jinyu Gao, Guanghan Li","doi":"10.1007/s12220-024-01759-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01759-7","url":null,"abstract":"<p>In this paper, we introduce an anisotropic geometric quantity <span>(mathbb {W}_{p,q;k} )</span> which involves the weighted integral of <i>k</i>-th elementary symmetric function. We first show the monotonicity of <span>({mathbb {W}}_{p,1;k})</span> and <span>({mathbb {W}}_{0,q;k})</span> along a class of inverse anisotropic curvature flows, and then prove the generalization of anisotropic Alexandrov–Fenchel type inequalities. On the other hand, an extension of anisotropic Hsiung–Minkowski formula is derived. Therefore, we at last obtain an extension of the Alexandrov–Fenchel type inequality, which involve the general <span>(mathbb {W}_{p,q;k})</span>. In terms of the above inequalities, we have also demonstrated some other meaningful conclusions on convex body geometry, such as generalized <span>(L^p)</span>-Minkowski inequality and estimates of anisotropic <i>p</i>-affine surface area.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190151","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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":null,"pages":null},"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
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":null,"pages":null},"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
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