Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of equations with unbounded terms

Abstract

We prove two versions of the classical Aleksandrov-Bakelman-Pucci (ABP) maximum principle using norms over contact sets for Lp-viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations F(x,u,Du,D2u)=f(x) in Ω⊂Rn, with measurable and unbounded terms. More precisely, we assume that the structural coefficient function γ (corresponding to the drift coefficient for linear equations) is in Ln(Ω). Such a result was previously known for Ln-viscosity solutions only when γ was bounded. We use the ABP maximum principle to prove pointwise properties of Lp-viscosity solutions for equations with unbounded γ and extend the theory of Lp-viscosity solutions to the case of equations with γ∈Ln(Ω). We use a recent generalization of ABP maximum principle by Krylov [28].