Alexandroff–Bakelman–Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

Abstract

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman–Isaacs equations from stochastic control problems. We establish an Alexandroff–Bakelman–Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.