A power penalty approach to a mixed quasilinear elliptic complementarity problem
type: Item
instance of: scholarly article
title: A power penalty approach to a mixed quasilinear elliptic complementarity problem
zbMATH Open document ID: 1490.90287
author: Y. Duan
author: Song Wang
author: Yuying Zhou
published in: Journal of Global Optimization
publication date: 2021-11-26T00:00:00Z
review text: In the paper under review, the authors consider and study the following mixed complementarity problem for the quasilinear elliptic type: \textbf{(MCP)} \ \ \ Find a pair \((u,\mu)\), such that the following conditions hold: \begin{gather*} Z(x)=-\operatorname{div}H(\nabla u)+G(u)-f(x)+\mu (x)=0,\\ \mu (x)\geq 0,\ u(x)-u^*(x)\leq 0,\ \mu(x)(u(x)-u^*(x))=0,\\ Z(x)\geq0,\ u_*(x)-u(x)\leq0,\ Z(x)(u_*(x)-u(x))=0, \end{gather*} for \(x\in \Omega\) a.e with the boundary condition: \(u(x)=0\), a.e. \(x\in \partial \Omega\), where is a bounded domain of \( \mathbb{R} ^{N}\), with piecewise smooth boundary \(\partial \Omega,\operatorname{div}\) denotes the usual divergence operator, \(u^{\ast},u_{\ast}:\Omega \rightarrow \mathbb{R}\) are given functions, \(f:\Omega \rightarrow \mathbb{R}\), \(H: \mathbb{R} ^{N}\rightarrow \mathbb{R} ^{N}\), \(\mu:\Omega \rightarrow \mathbb{R}\) and \(G: \mathbb{R} \rightarrow \mathbb{R}\) are functions satisfying in some given conditions. The problem (MCP) can be related to the necessary optimality conditions of some optimization problems. In addition, the problem (MCP) is equivalent to a double obstacle quasilinear elliptic variational inequality problem (denoted by VIP). The authors consider and study the following (VIP) problem related to (MCP): \textbf{(VIP)} \ \ \ Find \(u\in K:=\left \{ v\in X:u_{\ast}\leq v\leq u^{\ast }\right \}\) such that \[ (J(u),v-u)\geq \left \langle f,v-u\right \rangle ,\ \ \ \forall v\in K \] where \(X=W_{0}^{1,p},\ 2<p<\infty,u_{\ast}\leq0\leq u^{\ast}\)\ and \(J(u)\) is given by \(J(u)=-\operatorname{div}H(\nabla u)+G(u)\). The authors show that the (MCP) is equivalent to the problem (VIP). Now let \(k\geq1\), \(e\in \mathbb{R}\), \((e)_{+}=\max (e,0),(e)_{-} =\min \left \{ 0,e\right \}\), \(\chi(e)=\chi (\mathbb{R} _{+})-\chi (\mathbb{R} _{-})\) and set \(\varphi (u)=(u-u_{\ast})_{-}+(u-u^{\ast})_{+}\). For \(0<\varepsilon \ll1\), define \(\sigma_{\varepsilon}(u):=(\left \vert \varphi (u)\right \vert +\varep…
reviewed by: Mohammad Hossein Alizadeh
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 7433943
zbMATH Keywords: penalty approximation method
zbMATH Keywords: double obstacle
zbMATH Keywords: mixed complementarity problem
zbMATH Keywords: quasilinear elliptic variational inequality
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
full work available at URL: s10898-021-01000-7
OpenAlex ID: W3133383210
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DOI: 10.1007/S10898-021-01000-7
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description: scientific article; zbMATH DE number 7433943