{"title":"非局部梯度为零的非常数函数及其在非局部新曼类问题中的作用","authors":"","doi":"10.1016/j.na.2024.113642","DOIUrl":null,"url":null,"abstract":"<div><p>This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via <span><math><mi>Γ</mi></math></span>-convergence as the fractional parameter tends to 1.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001615/pdfft?md5=553f4dd248401bdbae37ffd61c633f93&pid=1-s2.0-S0362546X24001615-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113642\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via <span><math><mi>Γ</mi></math></span>-convergence as the fractional parameter tends to 1.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001615/pdfft?md5=553f4dd248401bdbae37ffd61c633f93&pid=1-s2.0-S0362546X24001615-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001615\",\"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":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001615","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems
This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via -convergence as the fractional parameter tends to 1.
期刊介绍:
Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.