Maximum principle for Pucci equations with sublinear growth in \(D u\) and its applications (Q2011499)

In this paper, the authors study the Dirichlet boundary value problems for Pucci extremal equations with sublinear growth terms in the first derivatives: \[ \mathcal P^{\pm}(D^2u)+\gamma|Du|+\mu|Du|^m=f\text{ in }\Omega \] where \(\Omega\in\mathbb R^n\), \(0<m<1\), and \(\gamma,\mu,f:\Omega\to\mathbb R\) are in \(L^p\) spaces with appropriate powers \(p\). The paper is organized in 5 sections. After an introduction, in Section 1, in Section 2, the authors introduce notations and recall some known results. Section 3 is devoted to showing the existence of \(L^p\)-strong solutions. Using the existence result, the authors show the ABP (Alexandrov-Bakelman-Pucci) maximum principle for \(L^p\)-viscosity solutions. Section 4 is dedicated to Harnack inequality. In Section 5, as an application, the authors show the continuity of viscosity solutions of \(\alpha\)-Poisson type equation for \(1<\alpha<2\).