{"title":"Hamilton–Jacobi方程及其哈密顿量连续依赖于未知的Lipschitz","authors":"H. Ishii, Kaizhi Wang, Lin Wang, Jun Yan","doi":"10.1080/03605302.2021.1983598","DOIUrl":null,"url":null,"abstract":"Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":"47 1","pages":"417 - 452"},"PeriodicalIF":2.1000,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown\",\"authors\":\"H. Ishii, Kaizhi Wang, Lin Wang, Jun Yan\",\"doi\":\"10.1080/03605302.2021.1983598\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.\",\"PeriodicalId\":50657,\"journal\":{\"name\":\"Communications in Partial Differential Equations\",\"volume\":\"47 1\",\"pages\":\"417 - 452\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2021-08-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Partial Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/03605302.2021.1983598\",\"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/10.1080/03605302.2021.1983598","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown
Abstract We study the Hamilton–Jacobi equations in M and in where the Hamiltonian depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.
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.