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Minimal cluster computation for four planar regions with the same area 具有相同面积的四个平面区域的最小聚类计算
Geometric Flows Pub Date : 2018-04-01 DOI: 10.1515/geofl-2018-0007
Emanuele Paolini, A. Tamagnini
{"title":"Minimal cluster computation for four planar regions with the same area","authors":"Emanuele Paolini, A. Tamagnini","doi":"10.1515/geofl-2018-0007","DOIUrl":"https://doi.org/10.1515/geofl-2018-0007","url":null,"abstract":"Abstract The topology of a minimal cluster of four planar regions with equal areas and smallest possible perimeter was found in [9]. Here we describe the computation used to check that the symmetric cluster with the given topology is indeed the unique minimal cluster.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"42 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116055794","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: 4
The elastic flow of curves on the sphere 曲线在球体上的弹性流动
Geometric Flows Pub Date : 2018-03-24 DOI: 10.1515/geofl-2018-0001
A. Dall'Acqua, Tim Laux, Lin, Paola Pozzi, A. Spener
{"title":"The elastic flow of curves on the sphere","authors":"A. Dall'Acqua, Tim Laux, Lin, Paola Pozzi, A. Spener","doi":"10.1515/geofl-2018-0001","DOIUrl":"https://doi.org/10.1515/geofl-2018-0001","url":null,"abstract":"Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131925170","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: 19
Discretization and Approximation of Surfaces Using Varifolds 曲面的离散化与近似
Geometric Flows Pub Date : 2018-03-01 DOI: 10.1515/geofl-2018-0004
Blanche Buet, G. P. Leonardi, S. Masnou
{"title":"Discretization and Approximation of Surfaces Using Varifolds","authors":"Blanche Buet, G. P. Leonardi, S. Masnou","doi":"10.1515/geofl-2018-0004","DOIUrl":"https://doi.org/10.1515/geofl-2018-0004","url":null,"abstract":"Abstract We present some recent results on the possibility of extending the theory of varifolds to the realm of discrete surfaces of any dimension and codimension, for which robust notions of approximate curvatures, also allowing for singularities, can be defined. This framework has applications to discrete and computational geometry, as well as to geometric variational problems in discrete settings. We finally show some numerical tests on point clouds that support and confirm our theoretical findings.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129496290","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: 13
Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2 基于二阶演化方程的不可扩展端部夹紧弹性曲线的演化
Geometric Flows Pub Date : 2018-03-01 DOI: 10.1515/geofl-2018-0002
Chun-Chi Lin, Yang-Kai Lue
{"title":"Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2","authors":"Chun-Chi Lin, Yang-Kai Lue","doi":"10.1515/geofl-2018-0002","DOIUrl":"https://doi.org/10.1515/geofl-2018-0002","url":null,"abstract":"Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130927213","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: 2
Convex relaxation and variational approximation of the Steiner problem: theory and numerics 斯坦纳问题的凸松弛与变分逼近:理论与数值
Geometric Flows Pub Date : 2018-03-01 DOI: 10.1515/geofl-2018-0003
M. Bonafini
{"title":"Convex relaxation and variational approximation of the Steiner problem: theory and numerics","authors":"M. Bonafini","doi":"10.1515/geofl-2018-0003","DOIUrl":"https://doi.org/10.1515/geofl-2018-0003","url":null,"abstract":"Abstract We survey some recent results on convex relaxations and a variational approximation for the classical Euclidean Steiner tree problem and we see how these new perspectives can lead to effective numerical schemes for the identification of Steiner minimal trees.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115118762","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: 8
On the Classification of Networks Self–Similarly Moving by Curvature 曲率自相似移动网络的分类
Geometric Flows Pub Date : 2017-12-20 DOI: 10.1515/geofl-2017-0006
P. Baldi, E. Haus, C. Mantegazza
{"title":"On the Classification of Networks Self–Similarly Moving by Curvature","authors":"P. Baldi, E. Haus, C. Mantegazza","doi":"10.1515/geofl-2017-0006","DOIUrl":"https://doi.org/10.1515/geofl-2017-0006","url":null,"abstract":"Abstract We give an overview of the classification of networks in the plane with at most two triple junctions with the property that under the motion by curvature they are self-similarly shrinking. After the contributions in [7, 9, 20], such a classification was completed in the recent work in [4] (see also [3]), proving that there are no self-shrinking networks homeomorphic to the Greek “theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees. We present the main geometric ideas behind the work [4]. We also briefly introduce our work in progress in the higher-dimensional case of networks of surfaces in R3.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125370617","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: 4
Some minimization problems for planar networks of elastic curves 平面弹性曲线网络的若干最小化问题
Geometric Flows Pub Date : 2017-10-27 DOI: 10.1515/geofl-2017-0005
A. Dall'Acqua, Alessandra Pluda
{"title":"Some minimization problems for planar networks of elastic curves","authors":"A. Dall'Acqua, Alessandra Pluda","doi":"10.1515/geofl-2017-0005","DOIUrl":"https://doi.org/10.1515/geofl-2017-0005","url":null,"abstract":"Abstract In this note we announce some results that will appear in [6] on the minimization of the functional F(Γ) = ∫Γk2 + 1 ds, where Γ is a network of three curves with fixed equal angles at the two junctions. The informal description of the results is accompanied by a partial review of the theory of elasticae and a diffuse discussion about the onset of interesting variants of the original problem passing from curves to networks. The considered energy functional F is given by the elastic energy and a term that penalize the total length of the network.We will show that penalizing the length is tantamount to fix it. The paper is concluded with the explicit computation of the penalized elastic energy of the “Figure Eight”, namely the unique closed elastica with self-intersections (see Figure 1).","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126556528","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: 13
Covers, soap films and BV functions 盖子，肥皂膜和BV功能
Geometric Flows Pub Date : 2017-10-03 DOI: 10.1515/geofl-2018-0005
G. Bellettini, M. Paolini, F. Pasquarelli, G. Scianna
{"title":"Covers, soap films and BV functions","authors":"G. Bellettini, M. Paolini, F. Pasquarelli, G. Scianna","doi":"10.1515/geofl-2018-0005","DOIUrl":"https://doi.org/10.1515/geofl-2018-0005","url":null,"abstract":"Abstract In this paper we review the double covers method with constrained BV functions for solving the classical Plateau’s problem. Next, we carefully analyze some interesting examples of soap films compatible with covers of degree larger than two: in particular, the case of a soap film only partially wetting a space curve, a soap film spanning a cubical frame but having a large tunnel, a soap film that retracts to its boundary, and various soap films spanning an octahedral frame.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128371888","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: 2
Equilibrium shapes of charged droplets and related problems: (mostly) a review 带电液滴的平衡形状及相关问题:(主要)综述
Geometric Flows Pub Date : 2017-09-14 DOI: 10.1515/geofl-2017-0004
M. Goldman, B. Ruffini
{"title":"Equilibrium shapes of charged droplets and related problems: (mostly) a review","authors":"M. Goldman, B. Ruffini","doi":"10.1515/geofl-2017-0004","DOIUrl":"https://doi.org/10.1515/geofl-2017-0004","url":null,"abstract":"Abstract We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this setting.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127321570","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: 4
Isoperimetric problems for a nonlocal perimeter of Minkowski type Minkowski型非局部周长的等周问题
Geometric Flows Pub Date : 2017-01-27 DOI: 10.1515/geofl-2017-0003
A. Cesaroni, M. Novaga
{"title":"Isoperimetric problems for a nonlocal perimeter of Minkowski type","authors":"A. Cesaroni, M. Novaga","doi":"10.1515/geofl-2017-0003","DOIUrl":"https://doi.org/10.1515/geofl-2017-0003","url":null,"abstract":"Abstract We show a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under volume and convexity constraints.We prove existence of minimizers, and we describe their shape as the volume tends to zero or to infinity.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"217 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130377053","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: 7
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