{"title":"具有可变指数的双相函数最小值的部分正则性","authors":"Atsushi Tachikawa","doi":"10.1007/s00030-023-00919-y","DOIUrl":null,"url":null,"abstract":"<p>The aim of this article is to study partial regularity of a minimizer <span>\\(\\varvec{u:\\Omega \\subset \\mathbb {R}^n \\rightarrow \\mathbb {R}^N}\\)</span> for a double phase functional with variable exponents: </p><span>$$\\begin{aligned} \\varvec{\\int \\left( \\vert Du\\vert _A^{p(x)} + a(x) { \\vert } Du{ \\vert } _A^{q(x)}\\right) dx,} \\end{aligned}$$</span><p>where <span>\\(\\varvec{{ \\vert } \\cdot { \\vert }_A}\\)</span> stands for the norm deduced from a positive definite sufficiently continuous tensor field <span>\\(\\varvec{A:=\\big (A_{\\alpha \\beta }^{ij} (x,u)\\big )~~((x,u) \\in \\Omega \\times \\mathbb {R}^N)}\\)</span>. We show that a minimizer <span>\\(\\varvec{u}\\)</span> is in the class <span>\\(\\varvec{C^{1,\\gamma }(\\Omega _1;\\mathbb {R}^N)}\\)</span> for some constant <span>\\(\\varvec{\\gamma \\in (0,1)}\\)</span> and open subset <span>\\(\\varvec{\\Omega _1 \\subset \\Omega }\\)</span>. We obtain also an estimate for the Hausdorff dimension of <span>\\(\\varvec{\\Omega \\setminus \\Omega _1}\\)</span>.</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Partial regularity of minimizers for double phase functionals with variable exponents\",\"authors\":\"Atsushi Tachikawa\",\"doi\":\"10.1007/s00030-023-00919-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of this article is to study partial regularity of a minimizer <span>\\\\(\\\\varvec{u:\\\\Omega \\\\subset \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^N}\\\\)</span> for a double phase functional with variable exponents: </p><span>$$\\\\begin{aligned} \\\\varvec{\\\\int \\\\left( \\\\vert Du\\\\vert _A^{p(x)} + a(x) { \\\\vert } Du{ \\\\vert } _A^{q(x)}\\\\right) dx,} \\\\end{aligned}$$</span><p>where <span>\\\\(\\\\varvec{{ \\\\vert } \\\\cdot { \\\\vert }_A}\\\\)</span> stands for the norm deduced from a positive definite sufficiently continuous tensor field <span>\\\\(\\\\varvec{A:=\\\\big (A_{\\\\alpha \\\\beta }^{ij} (x,u)\\\\big )~~((x,u) \\\\in \\\\Omega \\\\times \\\\mathbb {R}^N)}\\\\)</span>. We show that a minimizer <span>\\\\(\\\\varvec{u}\\\\)</span> is in the class <span>\\\\(\\\\varvec{C^{1,\\\\gamma }(\\\\Omega _1;\\\\mathbb {R}^N)}\\\\)</span> for some constant <span>\\\\(\\\\varvec{\\\\gamma \\\\in (0,1)}\\\\)</span> and open subset <span>\\\\(\\\\varvec{\\\\Omega _1 \\\\subset \\\\Omega }\\\\)</span>. We obtain also an estimate for the Hausdorff dimension of <span>\\\\(\\\\varvec{\\\\Omega \\\\setminus \\\\Omega _1}\\\\)</span>.</p>\",\"PeriodicalId\":501665,\"journal\":{\"name\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-02-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00030-023-00919-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-023-00919-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Partial regularity of minimizers for double phase functionals with variable exponents
The aim of this article is to study partial regularity of a minimizer \(\varvec{u:\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N}\) for a double phase functional with variable exponents:
where \(\varvec{{ \vert } \cdot { \vert }_A}\) stands for the norm deduced from a positive definite sufficiently continuous tensor field \(\varvec{A:=\big (A_{\alpha \beta }^{ij} (x,u)\big )~~((x,u) \in \Omega \times \mathbb {R}^N)}\). We show that a minimizer \(\varvec{u}\) is in the class \(\varvec{C^{1,\gamma }(\Omega _1;\mathbb {R}^N)}\) for some constant \(\varvec{\gamma \in (0,1)}\) and open subset \(\varvec{\Omega _1 \subset \Omega }\). We obtain also an estimate for the Hausdorff dimension of \(\varvec{\Omega \setminus \Omega _1}\).