## Abstract

We prove two versions of the classical Aleksandrov-Bakelman-Pucci (ABP) maximum principle using norms over contact sets for L^{p}-viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations F(x,u,Du,D^{2}u)=f(x) in Ω⊂R^{n}, 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 L^{n}(Ω). Such a result was previously known for L^{n}-viscosity solutions only when γ was bounded. We use the ABP maximum principle to prove pointwise properties of L^{p}-viscosity solutions for equations with unbounded γ and extend the theory of L^{p}-viscosity solutions to the case of equations with γ∈L^{n}(Ω). We use a recent generalization of ABP maximum principle by Krylov [28].

Original language | English |
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Pages (from-to) | 192-212 |

Number of pages | 21 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 168 |

DOIs | |

Publication status | Published - 2022 Dec |

## Keywords

- Aleksandrov-Bakelman-Pucci maximum principle
- Fully nonlinear uniformly elliptic equations
- L-viscosity solutions

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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