{"title":"k-Hessian 方程的边界正则性","authors":"You Li, Meng Ni Li, Yan Nan Liu","doi":"10.1007/s10114-023-0066-9","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we focus on the boundary regularity for a class of <i>k</i>-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the <i>k</i>-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the <i>k</i>-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Boundary Regularity for k-Hessian Equations\",\"authors\":\"You Li, Meng Ni Li, Yan Nan Liu\",\"doi\":\"10.1007/s10114-023-0066-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper we focus on the boundary regularity for a class of <i>k</i>-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the <i>k</i>-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the <i>k</i>-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-12-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-023-0066-9\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-0066-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this paper we focus on the boundary regularity for a class of k-Hessian equations which can be degenerate and (or) singular on the boundary of the domain. Motivated by the case of Monge–Ampère equations, we first construct sub-solutions, then apply the characteristic of the global Hölder continuity for convex functions, and finally use the maximum principle to obtain the boundary Hölder continuity for the solutions of the k-Hessian equations. However, finding such sub-solutions is very difficult due to the complexity of the k-Hessian operator. In particular, we employ the symmetric mean to overcome the difficulties.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.