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Comparison theorems on H-type sub-Riemannian manifolds. h型子黎曼流形的比较定理。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-05 DOI: 10.1007/s00526-025-02992-w
Fabrice Baudoin, Erlend Grong, Luca Rizzi, Sylvie Vega-Molino
{"title":"Comparison theorems on H-type sub-Riemannian manifolds.","authors":"Fabrice Baudoin, Erlend Grong, Luca Rizzi, Sylvie Vega-Molino","doi":"10.1007/s00526-025-02992-w","DOIUrl":"https://doi.org/10.1007/s00526-025-02992-w","url":null,"abstract":"<p><p>On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"143"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12053226/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143954637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Globally stable blowup profile for supercritical wave maps in all dimensions. 全维超临界波图的全局稳定爆破剖面。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-01-06 DOI: 10.1007/s00526-024-02901-7
Irfan Glogić
{"title":"Globally stable blowup profile for supercritical wave maps in all dimensions.","authors":"Irfan Glogić","doi":"10.1007/s00526-024-02901-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02901-7","url":null,"abstract":"<p><p>We consider wave maps from the <math><mrow><mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo>)</mo></mrow> </math> -dimensional Minkowski space into the <i>d</i>-sphere. It is known from the work of Bizoń and Biernat (Commun Math Phys 338(3): 1443-1450, 2015) that in the energy-supercritical case, i.e., for <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> , this model admits a closed-form corotational self-similar blowup solution. We show that this blowup profile is globally nonlinearly stable for all <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> , thereby verifying a perturbative version of the conjecture posed in Bizoń and Biernat (Commun Math Phys 338(3): 1443-1450, 2015) about the generic large data blowup behavior for this model. To accomplish this, we develop a novel stability analysis approach based on similarity variables posed on the whole space <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> . As a result, we draw a general road map for studying spatially global stability of self-similar blowup profiles for nonlinear wave equations in the radial case for arbitrary dimension <math><mrow><mi>d</mi> <mo>≥</mo> <mn>3</mn></mrow> </math> .</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 2","pages":"46"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11703941/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142945062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow. 光滑度量空间上的非线性快速扩散方程：Hamilton-Souplet-Zhang估计和Ricci-Perelman超流。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-02-06 DOI: 10.1007/s00526-025-02938-2
Ali Taheri, Vahideh Vahidifar
{"title":"The nonlinear fast diffusion equation on smooth metric measure spaces: Hamilton-Souplet-Zhang estimates and a Ricci-Perelman super flow.","authors":"Ali Taheri, Vahideh Vahidifar","doi":"10.1007/s00526-025-02938-2","DOIUrl":"https://doi.org/10.1007/s00526-025-02938-2","url":null,"abstract":"<p><p>This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the <i>f</i>-Laplacian. The gradient estimates of interest are of Hamilton-Souplet-Zhang or elliptic type and are established using different methods and techniques. Various implications, notably to parabolic Liouville type results and characterisation of ancient solutions are given. The problem is considered in the general setting where the metric and potential evolve under a super flow involving the Bakry-Émery <i>m</i>-Ricci curvature tensor. The curious interplay between geometry, nonlinearity, and evolution - and their intricate roles in the estimates and the maximum exponent range of fast diffusion - is at the core of the investigation.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 3","pages":"81"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11976860/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143977555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness. 不同密度演化表面上两相流的扩散界面模型：全局适定性。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-04 DOI: 10.1007/s00526-025-03001-w
Helmut Abels, Harald Garcke, Andrea Poiatti
{"title":"Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness.","authors":"Helmut Abels, Harald Garcke, Andrea Poiatti","doi":"10.1007/s00526-025-03001-w","DOIUrl":"https://doi.org/10.1007/s00526-025-03001-w","url":null,"abstract":"<p><p>We show global in time existence and uniqueness on any finite time interval of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"141"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12050238/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143978988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimal metrics for the first curl eigenvalue on 3-manifolds. 3流形上第一旋度特征值的最优度量。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-05 DOI: 10.1007/s00526-025-02995-7
Alberto Enciso, Wadim Gerner, Daniel Peralta-Salas
{"title":"Optimal metrics for the first curl eigenvalue on 3-manifolds.","authors":"Alberto Enciso, Wadim Gerner, Daniel Peralta-Salas","doi":"10.1007/s00526-025-02995-7","DOIUrl":"https://doi.org/10.1007/s00526-025-02995-7","url":null,"abstract":"<p><p>In this article we analyze the spectral properties of the curl operator on closed Riemannian 3-manifolds. Specifically, we study metrics that are optimal in the sense that they minimize the first curl eigenvalue among any other metric of the same volume in the same conformal class. We establish a connection between optimal metrics and the existence of minimizers for the <math><msup><mi>L</mi> <mfrac><mn>3</mn> <mn>2</mn></mfrac> </msup> </math> -norm in a fixed helicity class, which is exploited to obtain necessary and sufficient conditions for a metric to be locally optimal. As a consequence, our main result is that we prove that <math> <msup><mrow><mi>S</mi></mrow> <mn>3</mn></msup> </math> and <math><mrow><mi>R</mi> <msup><mrow><mi>P</mi></mrow> <mn>3</mn></msup> </mrow> </math> endowed with the round metric are <math><msup><mi>C</mi> <mn>1</mn></msup> </math> -local minimizers for the first curl eigenvalue (in its conformal and volume class). The connection between the curl operator and the Hodge Laplacian allows us to infer that the canonical metrics of <math> <msup><mrow><mi>S</mi></mrow> <mn>3</mn></msup> </math> and <math><mrow><mi>R</mi> <msup><mrow><mi>P</mi></mrow> <mn>3</mn></msup> </mrow> </math> are locally optimal for the first eigenvalue of the Hodge Laplacian on coexact 1-forms. This is in strong contrast to what happens in four dimensions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"146"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12052916/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143981510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices 通过基本矩阵的特征值求欧几里得球中最小 Legendrian 子满足的刚性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-19 DOI: 10.1007/s00526-024-02822-5
Pei-Yi Wu, Ling Yang
{"title":"The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices","authors":"Pei-Yi Wu, Ling Yang","doi":"10.1007/s00526-024-02822-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02822-5","url":null,"abstract":"<p>In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu’s inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"37 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250268","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
Isoperimetry and the properness of weak inverse mean curvature flow 等压法和弱反平均曲率流的适当性
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-18 DOI: 10.1007/s00526-024-02832-3
Kai Xu
{"title":"Isoperimetry and the properness of weak inverse mean curvature flow","authors":"Kai Xu","doi":"10.1007/s00526-024-02832-3","DOIUrl":"https://doi.org/10.1007/s00526-024-02832-3","url":null,"abstract":"<p>We prove a new existence theorem for proper solutions of Huisken and Ilmanen’s weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250269","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 the $$L^{p}$$ dual Minkowski problem for $$-1<0$$ 关于$$-1<0$$的$$L^{p}$$对偶闵科夫斯基问题
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-17 DOI: 10.1007/s00526-024-02806-5
Stephanie Mui
{"title":"On the $$L^{p}$$ dual Minkowski problem for $$-1<0$$","authors":"Stephanie Mui","doi":"10.1007/s00526-024-02806-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02806-5","url":null,"abstract":"<p>The <span>(L^{p})</span> dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the <span>(L^{p})</span> dual Minkowski problem, asks about existence of a convex body with prescribed <span>(L^{p})</span> dual curvature measure. This question unifies the previously disjoint <span>(L^{p})</span> Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the <span>(L^{p})</span> dual Minkowski problem for the case of <span>(q&lt;p+1)</span>, <span>(-1&lt;p&lt;0)</span>, and <span>(pne q)</span> for even measures.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"11 14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250272","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
Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion 作为具有非线性扩散的凯勒-西格尔系统奇异极限的赫勒-肖流
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-17 DOI: 10.1007/s00526-024-02826-1
Antoine Mellet
{"title":"Hele-Shaw flow as a singular limit of a Keller-Segel system with nonlinear diffusion","authors":"Antoine Mellet","doi":"10.1007/s00526-024-02826-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02826-1","url":null,"abstract":"<p>We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the <span>(Gamma )</span> convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the congested PKS model (which includes a density constraint <span>(rho le 1)</span> and a pressure term): It shows that the congestion constraint is not necessary to observe phase separation and surface tension phenomena.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"15 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250270","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
Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms 带有加权非局部源和梯度吸收项的准线性椭圆微分不等式的利乌维尔类型定理
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-16 DOI: 10.1007/s00526-024-02821-6
Ye Du, Zhong Bo Fang
{"title":"Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms","authors":"Ye Du, Zhong Bo Fang","doi":"10.1007/s00526-024-02821-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02821-6","url":null,"abstract":"<p>This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a strongly <i>p</i>-coercive elliptic differential inequality with weighted nonlocal source and gradient absorption terms in the whole space. Under the condition that the positive weight in the absorption term is either a sufficiently small constant or more general, we establish new Liouville type results containing the critical case. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250274","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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