On very weak solutions to a class of double obstacle problems
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 \(\mathcal{K}_{\varphi,\psi}^{r,\theta}\) there exists a constant \(C>0\) such that, for \(F,G\in L^{\frac{r}{p-1}}(\Omega)\), one has \[ \|u_F-u_G\|_{W_0^{1,r}(\Omega)}\leq C\|F-G\|^{\frac{1}{p-1}}_{L^{\frac{r}{p-1}}}. \] The proof of the existence of \(u_F\) is based on an approximation argument. Finally, a stability property for \(u_F\) with respect to variation of the obstacle functions is also established.
scientific article; zbMATH DE number 6156160