ABP Estimates for Viscosity Solutions of Elliptic Equations with Unbounded Coefficients in Large Domains

Abstract In this paper we consider fully nonlinear second-order elliptic partial differential equations, with measurable, possibly singular and unbounded lower order coefficients, in unbounded domains $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n . We prove the Alexandroff–Bakelman–Pucci maximum principle for $$L^p-$$ L p - viscosity solutions under suitable assumptions on the geometry of $$\Omega $$ Ω , satisfied by a class of very large unbounded domains including the complement of hypersurfaces, such as the cut plane in dimension $$n=2$$ n = 2 and the closure of $$(n-1)$$ ( n - 1 ) -dimensional open convex cones in higher dimensions, as well as their subdomains.