{"title":"The geometry of planar p-harmonic mappings: convexity, level curves and the isoperimetric inequality","authors":"Tomasz Adamowicz","doi":"10.2422/2036-2145.201201_010","DOIUrl":null,"url":null,"abstract":"We discuss various representations of planar p-harmonic systems of \nequations and their solutions. For coordinate functions of p-harmonic maps we \nanalyze signs of their Hessians, the Gauss curvature of p-harmonic surfaces, the \nlength of level curves as well as we discuss curves of steepest descent. The \nisoperimetric inequality for the level curves of coordinate functions of planar pharmonic \nmaps is proven. Our main techniques involve relations between quasiregular \nmaps and planar PDEs. We generalize some results due to P. Lindqvist, \nG. Alessandrini, G. Talenti and P. Laurence.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"1 1","pages":"263-292"},"PeriodicalIF":1.2000,"publicationDate":"2013-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2422/2036-2145.201201_010","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 9
Abstract
We discuss various representations of planar p-harmonic systems of
equations and their solutions. For coordinate functions of p-harmonic maps we
analyze signs of their Hessians, the Gauss curvature of p-harmonic surfaces, the
length of level curves as well as we discuss curves of steepest descent. The
isoperimetric inequality for the level curves of coordinate functions of planar pharmonic
maps is proven. Our main techniques involve relations between quasiregular
maps and planar PDEs. We generalize some results due to P. Lindqvist,
G. Alessandrini, G. Talenti and P. Laurence.
讨论了平面p调和方程组的各种表示形式及其解。对于p调和映射的坐标函数，我们分析了它们的Hessians符号、p调和曲面的高斯曲率、水平曲线的长度以及最陡下降曲线。证明了平面谐波映射坐标函数等距曲线的等距不等式。我们的主要技术涉及拟正则映射与平面偏微分方程之间的关系。我们推广了P. Lindqvist, G. Alessandrini, G. Talenti和P. Laurence的一些结果。
期刊介绍：
The Annals of the Normale Superiore di Pisa, Science Class, publishes papers that contribute to the development of Mathematics both from the theoretical and the applied point of view. Research papers or papers of expository type are considered for publication.
The Annals of the Normale Scuola di Pisa - Science Class is published quarterly
Soft cover, 17x24
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