On the Aleksandrov-Bakel'man-Pucci estimate for some elliptic and parabolic nonlinear operators
The authors prove the Aleksandrov-Bakel'man-Pucci estimate for (possibly degenerate) nonlinear elliptic and parabolic equations of the form \[ -{\text{div}} (F(\nabla u(x))) = f(x) \quad {\text{in}}\,\Omega \subset \mathbb{R}^{n} \] and \[ u_{t}(x,t)-{\text{div}}(F(\nabla u(x,t))) = f(x,t) \quad {\text{in}}\,Q\subset \mathbb{R}^{n+1} \] for \(F\) a \({\mathcal{C}^1}\) monotone field under some suitable conditions. Examples of applications such as the \(p\)-Laplacian and the Mean Curvature Flow are considered, as well as extensions of the general results to equations that are not in divergence form, such as the \(m\)-curvature flow.
scientific article; zbMATH DE number 6101966
scientific article