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A dynamical approach to the variational inequality on modified elastic graphs 修正弹性图上变分不等式的一种动力学方法
Geometric Flows Pub Date : 2020-01-01 DOI: 10.1515/geofl-2020-0100
S. Okabe, Kensuke Yoshizawa
{"title":"A dynamical approach to the variational inequality on modified elastic graphs","authors":"S. Okabe, Kensuke Yoshizawa","doi":"10.1515/geofl-2020-0100","DOIUrl":"https://doi.org/10.1515/geofl-2020-0100","url":null,"abstract":"Abstract We consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122982211","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: 6
Simply connected translating solitons contained in slabs 平板中包含的单连通平移孤子
Geometric Flows Pub Date : 2019-12-28 DOI: 10.1515/GEOFL-2020-0101
Francesco Chini
{"title":"Simply connected translating solitons contained in slabs","authors":"Francesco Chini","doi":"10.1515/GEOFL-2020-0101","DOIUrl":"https://doi.org/10.1515/GEOFL-2020-0101","url":null,"abstract":"In this work we show that 2-dimensional, simply connected, translating solitons of the mean curvature flow embedded in a slab of ℝ3 with entropy strictly less than 3 must be mean convex and thus, thanks to a result by Spruck and Xiao are convex. Recently, such 2-dimensional convex translating solitons have been completely classified, up to an ambient isometry, as vertical planes, (tilted) grim reaper cylinders, Δ-wings and bowl translater. These are all contained in a slab, except for the rotationally symmetric bowl translater. New examples by Ho man, Martín and White show that the bound on the entropy is necessary.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134115445","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: 5
On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions 高维空间中具有三重结的表面扩散流
Geometric Flows Pub Date : 2019-07-26 DOI: 10.1515/geofl-2020-0001
H. Garcke, M. Gößwein
{"title":"On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions","authors":"H. Garcke, M. Gößwein","doi":"10.1515/geofl-2020-0001","DOIUrl":"https://doi.org/10.1515/geofl-2020-0001","url":null,"abstract":"Abstract We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128440256","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: 6
On the Convergence of the Elastic Flow in the Hyperbolic Plane 关于弹性流在双曲平面上的收敛性
Geometric Flows Pub Date : 2019-01-10 DOI: 10.1515/geofl-2020-0002
Marius Müller, A. Spener
{"title":"On the Convergence of the Elastic Flow in the Hyperbolic Plane","authors":"Marius Müller, A. Spener","doi":"10.1515/geofl-2020-0002","DOIUrl":"https://doi.org/10.1515/geofl-2020-0002","url":null,"abstract":"Abstract We examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127571602","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: 11
Prescribing scalar curvatures: non compactness versus critical points at infinity 规定标量曲率:非紧性与无穷远处的临界点
Geometric Flows Pub Date : 2019-01-01 DOI: 10.1515/geofl-2019-0004
Martin Gebhard Mayer
{"title":"Prescribing scalar curvatures: non compactness versus critical points at infinity","authors":"Martin Gebhard Mayer","doi":"10.1515/geofl-2019-0004","DOIUrl":"https://doi.org/10.1515/geofl-2019-0004","url":null,"abstract":"Abstract We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121508973","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: 5
Elastic flow of networks: long-time existence result 网络弹性流动:长期存在的结果
Geometric Flows Pub Date : 2019-01-01 DOI: 10.1515/geofl-2019-0005
A. Dall'Acqua, Chun-Chi Lin, Paola Pozzi
{"title":"Elastic flow of networks: long-time existence result","authors":"A. Dall'Acqua, Chun-Chi Lin, Paola Pozzi","doi":"10.1515/geofl-2019-0005","DOIUrl":"https://doi.org/10.1515/geofl-2019-0005","url":null,"abstract":"Abstract We provide a long-time existence and sub-convergence result for the elastic flow of a three-network in ℝn under some mild topological assumptions. The evolution is such that the sum of the elastic energies of the three curves plus their weighted lengths decrease in time. Natural boundary conditions are considered at the boundary of the curves and at the triple junction.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134511291","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: 14
On obstacle problem for mean curvature flow with driving force 带驱动力的平均曲率流的障碍问题
Geometric Flows Pub Date : 2018-10-04 DOI: 10.1515/GEOFL-2019-0002
Y. Giga, H. Tran, Longjie Zhang
{"title":"On obstacle problem for mean curvature flow with driving force","authors":"Y. Giga, H. Tran, Longjie Zhang","doi":"10.1515/GEOFL-2019-0002","DOIUrl":"https://doi.org/10.1515/GEOFL-2019-0002","url":null,"abstract":"Abstract In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116957180","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: 11
Rigidity of positively curved shrinking Ricci solitons in dimension four 四维正弯曲收缩Ricci孤子的刚性
Geometric Flows Pub Date : 2018-07-16 DOI: 10.1515/geofl-2019-0001
G. Catino
{"title":"Rigidity of positively curved shrinking Ricci solitons in dimension four","authors":"G. Catino","doi":"10.1515/geofl-2019-0001","DOIUrl":"https://doi.org/10.1515/geofl-2019-0001","url":null,"abstract":"Abstract We classify four-dimensional shrinking Ricci solitons satisfying Sec ≥ 124R$Sec, ge ,{1 over {24}}R$, where Sec and R denote the sectional and the scalar curvature, respectively. They are isometric to either ℝ4 (and quotients), 𝕊 4, ℝℙ4 or ℂ ℙ2 with their standard metrics.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128777619","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
Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow 多相平均曲率流数值格式分析的梯度流技术
Geometric Flows Pub Date : 2018-04-04 DOI: 10.1515/geofl-2018-0006
Tim Laux
{"title":"Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow","authors":"Tim Laux","doi":"10.1515/geofl-2018-0006","DOIUrl":"https://doi.org/10.1515/geofl-2018-0006","url":null,"abstract":"Abstract Several recent convergence results for numerical schemes for mean-curvature flow in particular in the multi-phase case with arbitrary surface tensions are discussed. The guiding principle of all these works is the gradient-flow structure of multi-phase mean-curvature flow which is explained in the general framework. For simplicity, the convergence results are presented in the simpler two-phase case.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130110628","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: 1
A metric-based approach to multiphase mean curvature flows with mobilities 多相平均曲率流动的一种基于度量的方法
Geometric Flows Pub Date : 2018-04-01 DOI: 10.1515/geofl-2018-0008
É. Bretin, A. Danescu, J. Penuelas, S. Masnou
{"title":"A metric-based approach to multiphase mean curvature flows with mobilities","authors":"É. Bretin, A. Danescu, J. Penuelas, S. Masnou","doi":"10.1515/geofl-2018-0008","DOIUrl":"https://doi.org/10.1515/geofl-2018-0008","url":null,"abstract":"Abstract We summarize in this note a few theoretical and numerical results described with more details in the forthcoming paper [5] devoted to a new phase field model for approximating mean curvature flows. Our work is motivated by the simulation of multiphase mean curvature flows with possibly inhomogeneous surface tensions and mobility coefficients. The key aspect of our model is to incorporate the mobilities not in the phase field energy (which is conventionally the case) but in the metric which determines the gradient flow. We show the consistency of such approach by a formal analysis of the sharp interface limit. We also propose an efficient numerical scheme which allows us to illustrate the advantages of the model on various examples, as the wetting of droplets on solid surfaces or the simulation of nanowires growth generated by the so-called vapor-liquid-solid method.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"253 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":"121885211","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: 1
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