A power penalty approach to a mixed quasilinear elliptic complementarity problem (Q2052401)

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 \(\sigma_{\varepsilon}(u):=(\left \vert \varphi (u)\right \vert +\varepsilon )^{\frac{1}{k}}\chi (\varphi (u))-\varepsilon^{\frac{1}{k}}\). They consider the following nonlinear penalized equation: \[ -\operatorname{div}H(\nabla u)+G(u)+\lambda \sigma_{\varepsilon}(u)=f\text{ a.e. }x\in \Omega, \] where \(\lambda>1\) is a parameter. The variational problem corresponding to this equation is \textbf{(\(\mathbf{P}_{\lambda}\))} \ \ \ Find \(u_{\lambda}\in X\) such that, for all \(v\in X\), \[ (J(u_{\lambda})+\lambda \sigma_{\varepsilon}(u_{\lambda}),v)=(f,u). \] The authors prove the existence and uniqueness of the solution for the above problem. Also, they show that the solution of the penalty equation converges to the unique solution of the (VIP) problem. In addition, a convergence analysis for the penalty method is established. Moreover, some numerical results regarding to the special case, \(H(\nabla u)=\left \vert \nabla u\right \vert ^{p-2}\) with \(p=2\),\ are presented.