A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems
Let \(u\in W^{1,p}(\Omega)\) be a solution of the variational inequality \[ \int_{\Omega}| Du|^{p-2} Du(Dv-Du)dx\geq \int_{\Omega}b(x,u,Du)(v-u)dx+\int_{\Omega}f(x)(Dv-Du)dx \] for all \(v\in \{W_ 0^{1,p}(\Omega)+u_ 0\}\), v(x)\(\geq \psi (x)\) a.e. in \(\Omega\), where \(u_ 0\in W^{1,p}(\Omega)\) and \(u_ 0(x)\geq \psi (x)\). Under appropriate conditions on b and f the author shows that \(u\in C_{loc}^{0,\alpha}(\Omega)\) when \(\psi =\in W_{loc}^{1,m}(\Omega)\) \((m>n)\) and \(u\in C_{loc}^{1,\alpha}(\Omega)\) when \(\psi \in W_{loc}^{1,\beta}(\Omega)\) \((\alpha >0)\) for some suitable \(\beta >0\).
scientific article; zbMATH DE number 4212573
scientific article