A Classical Analysis Counterpart of Viterbo's Symplectic Geometry Proof of ABP in the Plane

TL;DR

This paper presents a classical analysis proof of the ABP inequality in two dimensions inspired by symplectic geometry methods, and explores extensions to higher dimensions.

Contribution

It provides a novel classical analysis proof of ABP in 2D inspired by symplectic geometry, and discusses potential generalizations to higher dimensions.

Findings

Classical analysis proof of ABP in 2D without convexity or contact sets.

Extension of proof to remove compact support hypothesis and include boundary terms.

Discussion on challenges of extending proof to dimensions three and above.

Abstract

We first provide a classical analysis proof of a version of the Alexandroff-Bakelman-Pucci inequality (ABP) for compactly supported $C^2$ functions in dimension $2$, inspired by the symplectic geometry proof method of Viterbo, which avoids convexity or contact sets. We then show how the proof may be modified to remove the compact support hypothesis and recover the usual statement of ABP, which includes a boundary term. We also discuss the possibility (and difficulties) of extending a pure classical analysis proof to dimension $3$ and above.