On very weak solutions to a class of double obstacle problems
type: Item
instance of: scholarly article
title: On very weak solutions to a class of double obstacle problems
zbMATH Open document ID: 1270.35187
author: Gejun Bao
author: Guanfeng Li
author: Tingting Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-04-22T00:00:00Z
review text: Let \(\Omega\subset {\mathbb R}^n\) be a bounded regular domain. The authors study the uniqueness of very weak solutions of a double obstacle problem associated to the following quasilinear elliptic equation \[ -\mathrm{div}A(x,\nabla u)=-\mathrm{div}F(x), \;\;\;\;x\in \Omega, \tag{1} \] where \(F\in L^{\frac{r}{p-1}}(\Omega)\), with \(1<p,r<\infty\) and \(p-1<r\leq p\), and \(A:\Omega\times {\mathbb R}^n\rightarrow {\mathbb R}^n\) is a Carathéodory mapping such that, for fixed \(0<\alpha\leq \beta<\infty\) and for almost every \(x\in \Omega\), and all \(\xi,\zeta \in {\mathbb R}^n\), \(\lambda \in {\mathbb R}\) \(|A(x,\xi)-A(x,\zeta)|\leq \beta |\xi-\zeta|^{p-1}\); \(\langle A(x,\xi)-A(x,\zeta),\xi-\zeta \rangle \geq \alpha |\xi-\zeta|^2(|\xi|+|\zeta|)^{p-2}\) and \(A(x,\lambda \xi)=\lambda |\lambda|^{p-2}A(x,\xi)\). For each \(\theta \in W^{1,2}(\Omega)\) and \(\varphi,\psi: \Omega \rightarrow [-\infty,\infty]\), let \[ \mathcal{K}_{\varphi,\psi}^{r,\theta}=\{u\in W^{1,2}(\Omega): \varphi \leq u\leq \psi \text{ a.e. in }\Omega, \;u-\theta \in W_0^{1,r}(\Omega)\}. \] A function \(u\in \mathcal{K}_{\varphi,\psi}^{r,\theta}\) is called a very weak solution to the \(\mathcal{K}_{\varphi,\psi}^{r,\theta}\)-obstacle problem associated with equation \((1)\) if, for every \(v\in \mathcal{K}_{\varphi,\psi}^{r,\theta}\), \[ \int_\Omega \langle A(x,\nabla u),|\nabla(v-u)|^{r-p}\nabla (v-u) \rangle dx\geq \int_\Omega \langle A(x,\nabla u), H_{v,u}\rangle dx+\int_\Omega \langle F,\nabla \phi_{v,u}\rangle dx, \;\;\;\;\;\;\;(P_F) \] where \(\phi\in W_0^{1,\frac{r}{1+r-p}}(\Omega)\) and \(H\in L_0^{\frac{r}{1+r-p}}(\Omega)\) come from the Hodge decomposition of \(v-u\), that is \(|\nabla (v-u)|^{r-p}\nabla (v-u)=\nabla \phi_{v,u}+H_{v,u}\). The authors prove that, if \(r\) is sufficiently near to \(p\), then problem \((P_F)\) admits a very weak solution \(u_F\), which turns out to be a quasi-minimizer for the \(r\)-Dirichlet integral \(\int_\Omega |\nabla (\cdot)|^rdx\) in \(\mat…
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J87
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6156160
zbMATH Keywords: Hodge decomposition
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.01.065
OpenAlex ID: W1985468899
cites work: The obstacle problem revisited
cites work: Q4708169
cites work: Q4457306
cites work: Q4376706
cites work: On very weak solutions of degenerate \(p\)-harmonic equations
cites work: Q4693180
cites work: \(p\)-harmonic tensors and quasiregular mappings
cites work: Integrability and Removability Results for Quasiregular Mappings in High Dimensions.
cites work: Weak minima of variational integrals.
cites work: On very weak solutions of certain elliptic systems
cites work: Q4664466
cites work: Stability and higher integrability of derivatives of solutions in double obstacle problems
cites work: Q4256172
cites work: The existence of solutions to the nonhomogeneous \(\mathcal A\)-harmonic equation
cites work: Q3745373
cites work: Q4322914
cites work: On weakly A-harmonic tensors
cites work: Local boundedness results for very weak solutions of double obstacle problems
cites work: Q3801969
DOI: 10.1016/J.JMAA.2013.01.065
description: scientific article; zbMATH DE number 6156160