{"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":"89 1","pages":""},"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:
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
Targeted topics include:
in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis;
in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.