{"title":"∞-调和函数的流线服从逆平均曲率流","authors":"R. Moser","doi":"10.1080/03605302.2022.2109487","DOIUrl":null,"url":null,"abstract":"Abstract Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"2124 - 2145"},"PeriodicalIF":2.1000,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The streamlines of ∞-harmonic functions obey the inverse mean curvature flow\",\"authors\":\"R. Moser\",\"doi\":\"10.1080/03605302.2022.2109487\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.\",\"PeriodicalId\":50657,\"journal\":{\"name\":\"Communications in Partial Differential Equations\",\"volume\":\"47 1\",\"pages\":\"2124 - 2145\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2021-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/03605302.2022.2109487\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/03605302.2022.2109487","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The streamlines of ∞-harmonic functions obey the inverse mean curvature flow
Abstract Given an ∞-harmonic function on a domain consider the function If with and then it is easy to check that the streamlines of are the level sets of w and w solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, w is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of by p-harmonic functions, the use of conjugate -harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of arises as a by-product.
期刊介绍:
This journal aims to publish high quality papers concerning any theoretical aspect of partial differential equations, as well as its applications to other areas of mathematics. Suitability of any paper is at the discretion of the editors. We seek to present the most significant advances in this central field to a wide readership which includes researchers and graduate students in mathematics and the more mathematical aspects of physics and engineering.