{"title":"超立方体间最优映射的正则性","authors":"Shibing Chen, Jiakun Liu, Xu-Jia Wang","doi":"10.1515/ans-2023-0087","DOIUrl":null,"url":null,"abstract":"Abstract In this note, we establish the global <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,\\alpha } regularity for potential functions in optimal transportation between hypercubes in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\\mathbb{R}}}^{n} for <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> n\\ge 3 . When <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> n=2 , the result was proved by Jhaveri. The <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,\\alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"301 1","pages":"0"},"PeriodicalIF":2.1000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Regularity of optimal mapping between hypercubes\",\"authors\":\"Shibing Chen, Jiakun Liu, Xu-Jia Wang\",\"doi\":\"10.1515/ans-2023-0087\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this note, we establish the global <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,\\\\alpha } regularity for potential functions in optimal transportation between hypercubes in <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi mathvariant=\\\"double-struck\\\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\\\\mathbb{R}}}^{n} for <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> n\\\\ge 3 . When <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:math> n=2 , the result was proved by Jhaveri. The <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C}^{3,\\\\alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.\",\"PeriodicalId\":7191,\"journal\":{\"name\":\"Advanced Nonlinear Studies\",\"volume\":\"301 1\",\"pages\":\"0\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advanced Nonlinear Studies\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/ans-2023-0087\",\"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":"Advanced Nonlinear Studies","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/ans-2023-0087","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract In this note, we establish the global C3,α {C}^{3,\alpha } regularity for potential functions in optimal transportation between hypercubes in Rn {{\mathbb{R}}}^{n} for n≥3 n\ge 3 . When n=2 n=2 , the result was proved by Jhaveri. The C3,α {C}^{3,\alpha } regularity is also optimal due to a counterexample in the study by Jhaveri.
期刊介绍：
Advanced Nonlinear Studies is aimed at publishing papers on nonlinear problems, particulalry those involving Differential Equations, Dynamical Systems, and related areas. It will also publish novel and interesting applications of these areas to problems in engineering and the sciences. Papers submitted to this journal must contain original, timely, and significant results. Articles will generally, but not always, be published in the order when the final copies were received.
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