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The Alexandroff–Bakelman–Pucci estimate via positive drift

Antonio Vitolo

Vol. 5 (2023), No. 2, 261–283

A new Alexandroff–Bakelman–Pucci estimate is obtained for solutions of fully nonlinear uniform elliptic equations in any proper n-dimensional domain. The novelty, with respect to the existing literature, is that the estimate does not depend on the geometry of the domain and extends to solutions, vanishing at infinity, in arbitrary unbounded domains. No decay condition on the first-order coefficient is assumed, but instead a “positive” drift. The existence of solutions vanishing at infinity is also shown, based on the ABP estimates previously proved.

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fully nonlinear elliptic equations, ABP estimate, comparison principles, existence, regularity
Mathematical Subject Classification
Primary: 35B45
Secondary: 35J25, 35J60, 35D40, 35B50
Received: 22 August 2021
Revised: 9 September 2022
Accepted: 12 October 2022
Published: 26 June 2023
Antonio Vitolo
Dipartimento di Matematica
Università di Salerno