{"title":"Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems","authors":"Antonio Giuseppe Grimaldi, Erica Ipocoana","doi":"10.1051/cocv/2022050","DOIUrl":null,"url":null,"abstract":"We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \\begin {gather*} \\min \\biggl\\{ \\int_{\\Omega} F(x,w,Dw) d x \\ : \\ w \\in \\mathcal{K}_{\\psi}(\\Omega) \\biggr\\}, \\end {gather*} with $F$ double phase functional of the form \\begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \\end {equation*} where $\\Omega$ is a bounded open subset of $\\mathbb{R}^n$ , $\\psi \\in W^{1,p}(\\Omega)$ is a fixed function called \\textit { obstacle } and $\\mathcal{K}_{\\psi}(\\Omega)= \\{ w \\in W^{1,p}(\\Omega) : w \\geq \\psi \\ \\text{a.e. in} \\ \\Omega \\}$ is the class of admissible functions . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2022050","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 3
Abstract
We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin {gather*} \min \biggl\{ \int_{\Omega} F(x,w,Dw) d x \ : \ w \in \mathcal{K}_{\psi}(\Omega) \biggr\}, \end {gather*} with $F$ double phase functional of the form \begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end {equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^n$ , $\psi \in W^{1,p}(\Omega)$ is a fixed function called \textit { obstacle } and $\mathcal{K}_{\psi}(\Omega)= \{ w \in W^{1,p}(\Omega) : w \geq \psi \ \text{a.e. in} \ \Omega \}$ is the class of admissible functions . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .
期刊介绍:
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