{"title":"可逆性","authors":"Crista Arangala","doi":"10.1090/mmono/003/06","DOIUrl":null,"url":null,"abstract":". Let Ω, Ω ′ ⊂ R n be bounded domains and let f m : Ω → Ω ′ be a sequence of homeomorphisms with positive Jacobians J f m > 0 a.e. and prescribed Dirichlet boundary data. Let all f m satisfy the Lusin (N) condition and sup m R Ω ( | Df m | n − 1 + A ( | cof Df m | )+ ϕ ( J f )) < ∞ , where A and ϕ are positive convex functions. Let f be a weak limit of f m in W 1 ,n − 1 . Provided certain growth behaviour of A and ϕ , we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.","PeriodicalId":205141,"journal":{"name":"How to Pass the FRACP Written Examination","volume":"181 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"Invertibility\",\"authors\":\"Crista Arangala\",\"doi\":\"10.1090/mmono/003/06\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". Let Ω, Ω ′ ⊂ R n be bounded domains and let f m : Ω → Ω ′ be a sequence of homeomorphisms with positive Jacobians J f m > 0 a.e. and prescribed Dirichlet boundary data. Let all f m satisfy the Lusin (N) condition and sup m R Ω ( | Df m | n − 1 + A ( | cof Df m | )+ ϕ ( J f )) < ∞ , where A and ϕ are positive convex functions. Let f be a weak limit of f m in W 1 ,n − 1 . Provided certain growth behaviour of A and ϕ , we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.\",\"PeriodicalId\":205141,\"journal\":{\"name\":\"How to Pass the FRACP Written Examination\",\"volume\":\"181 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"How to Pass the FRACP Written Examination\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/mmono/003/06\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"How to Pass the FRACP Written Examination","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/mmono/003/06","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 9
摘要
． 设Ω， Ω’∧R n为有界域，设f m: Ω→Ω’为具有正雅可比矩阵J f m > 0 a.e.的同胚序列和规定的狄利克雷边界数据。令所有的f m满足Lusin (N)条件，并使m R Ω (| Df m | N−1 + A (| cof Df m |)+ φ (J f)) <∞，其中A和φ为正凸函数。设f是fm在w1,n−1中的弱极限。在给定A和φ的一定增长行为的情况下，我们证明了f满足Conti和De Lellis的(INV)条件和Lusin (N)条件，并且多凸能量是下半连续的。
. Let Ω, Ω ′ ⊂ R n be bounded domains and let f m : Ω → Ω ′ be a sequence of homeomorphisms with positive Jacobians J f m > 0 a.e. and prescribed Dirichlet boundary data. Let all f m satisfy the Lusin (N) condition and sup m R Ω ( | Df m | n − 1 + A ( | cof Df m | )+ ϕ ( J f )) < ∞ , where A and ϕ are positive convex functions. Let f be a weak limit of f m in W 1 ,n − 1 . Provided certain growth behaviour of A and ϕ , we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.
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