{"title":"On the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations","authors":"Xiaoliang Li, Cong Wang","doi":"10.1142/S0219199722500821","DOIUrl":null,"url":null,"abstract":". We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f ( λ ( D 2 u )) = g ( x ) , with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli–Nirenberg–Spruck [8], Trudinger [35] and many others, and there had been significant discussions on the solv- ability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on the Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure condi- tions as in [8, 35] but without requiring the concavity of f . The equations in our setting may embrace the well-known Monge–Amp`ere equations, Hessian equations and Hessian quotient equations as special cases.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/S0219199722500821","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 1
Abstract
. We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f ( λ ( D 2 u )) = g ( x ) , with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli–Nirenberg–Spruck [8], Trudinger [35] and many others, and there had been significant discussions on the solv- ability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on the Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure condi- tions as in [8, 35] but without requiring the concavity of f . The equations in our setting may embrace the well-known Monge–Amp`ere equations, Hessian equations and Hessian quotient equations as special cases.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.