Daniel Maienshein - Direct proof of the Alexandroff-Bakelman-Pucci inequality in the plane via Viterbo's symplectic geometry method

The classical proof of the Alexandroff-Bakelman-Pucci inequality (ABP) uses the area formula along with a lower bound on the measure of the image of the gradient by way of convex envelopes and contact sets. A surprising new proof of ABP emerged in 2000, when Viterbo deduced ABP for compactly supported functions using notions from symplectic geometry (namely, topological invariants of Lagrangian submanifolds of the cotangent bundle). In this talk, we present a direct proof of ABP in dimension 2 following Viterbo's approach, translated into the language of classical analysis. We mention how the proof can be modified to remove the compact support hypothesis. Finally, we discuss the possibility (and difficulties) of a pure classical analysis proof in dimension 3 and above. The content of this talk is based on collaboration with Juan Manfredi.