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Equivariant valuations on convex functions. 凸函数的等变赋值。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-09-08 DOI: 10.1007/s00526-025-03117-z
Georg C Hofstätter, Jonas Knoerr
{"title":"Equivariant valuations on convex functions.","authors":"Georg C Hofstätter, Jonas Knoerr","doi":"10.1007/s00526-025-03117-z","DOIUrl":"10.1007/s00526-025-03117-z","url":null,"abstract":"<p><p>We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 8","pages":"242"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12417309/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145039184","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":"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
Absolutely minimal semi-Lipschitz extensions. 绝对最小半利普希茨扩展。
IF 2 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-10-15 DOI: 10.1007/s00526-025-03169-1
Aris Daniilidis, Trí Minh Lê, Francisco M Venegas
{"title":"Absolutely minimal semi-Lipschitz extensions.","authors":"Aris Daniilidis, Trí Minh Lê, Francisco M Venegas","doi":"10.1007/s00526-025-03169-1","DOIUrl":"10.1007/s00526-025-03169-1","url":null,"abstract":"<p><p>The notion of quasi-metric space arises by revoking the symmetry from the definition of a distance. Semi-Lipschitz functions appear naturally as morphisms associated with the new structure. In this work, under suitable assumptions on the quasi-metric space (analogous to standard ones in the metric case), we establish existence of optimal (that is, absolutely minimal) extensions of real-valued semi-Lipschitz functions from a subset of the space to the whole space. This is done in two different ways: first, by adapting the Perron method from the classical setting to this asymmetric case, and second, by means of an iteration scheme for (an unbalanced version of) the tug-of-war game, initiating the algorithm from a McShane extension. This new iteration scheme provides, even in the symmetric case of a metric space, a constructive way of establishing existence of absolutely minimal Lipschitz extensions of real-valued Lipschitz functions.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 9","pages":"301"},"PeriodicalIF":2.0,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12528290/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145328269","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":"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
Γ -Limsup estimate for a nonlocal approximation of the Willmore functional. Γ -对Willmore泛函的非局部近似的limsup估计。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-30 DOI: 10.1007/s00526-025-03039-w
Hardy Chan, Mattia Freguglia, Marco Inversi
{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\"><ns0:math><ns0:mi>Γ</ns0:mi></ns0:math> -Limsup estimate for a nonlocal approximation of the Willmore functional.","authors":"Hardy Chan, Mattia Freguglia, Marco Inversi","doi":"10.1007/s00526-025-03039-w","DOIUrl":"https://doi.org/10.1007/s00526-025-03039-w","url":null,"abstract":"<p><p>We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation of the fractional Allen-Cahn energies, and we prove the corresponding <math><mi>Γ</mi></math> -limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen-Cahn energies.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 6","pages":"181"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12125130/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144198267","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
Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure. 不可压缩多孔介质方程及Stokes输运系统的能量结构稳定性分析。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-28 DOI: 10.1007/s00526-025-03029-y
Jaemin Park
{"title":"Stability analysis of the incompressible porous media equation and the Stokes transport system via energy structure.","authors":"Jaemin Park","doi":"10.1007/s00526-025-03029-y","DOIUrl":"https://doi.org/10.1007/s00526-025-03029-y","url":null,"abstract":"<p><p>In this paper, we revisit asymptotic stability for the two-dimensional incompressible porous media equation and the Stokes transport system in a periodic channel. It is well-known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small and smooth perturbations. We provide improvements in the regularity assumptions on the perturbation and in the convergence rate. Unlike the standard approach for stability analysis relying on linearized equations, we directly address the nonlinear problem by exploiting the energy structure of each system. While it is widely known that the potential energy is a Lyapunov functional in both systems, our key observation is that the second derivative of the potential energy reveals a (degenerate) coercive structure, which arises from the fact that the solution converges to the minimizer of the energy.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 5","pages":"169"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12119689/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144198266","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 Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities. 非局部波动方程的最优Runge近似和多齐次非线性的唯一确定。
IF 2.1 2区 数学
Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-10-10 DOI: 10.1007/s00526-025-03124-0
Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann
{"title":"Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities.","authors":"Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann","doi":"10.1007/s00526-025-03124-0","DOIUrl":"10.1007/s00526-025-03124-0","url":null,"abstract":"<p><p>The main purpose of this article is to establish the Runge-type approximation in <math> <mrow><msup><mi>L</mi> <mn>2</mn></msup> <mrow><mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup><mover><mi>H</mi> <mo>~</mo></mover> <mi>s</mi></msup> <mrow><mo>(</mo> <mi>Ω</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> </mrow> </math> for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calderón problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calderón problem of nonlocal wave equations with polyhomogeneous nonlinearities.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"64 9","pages":"280"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12513988/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145279017","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
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