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arXiv · 2509.10879

Alexandrov estimates for polynomial operators by determinant majorization

F. Reese Harvey·Kevin R. Payne
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Abstract

We obtain estimates on the supremum, infimum and oscillation of solutions for a wide class of inhomogeneous fully nonlinear elliptic equations on Euclidean domains where the differential operator is an I-central Garding-Dirichlet operator in the sense of Harvey-Lawson (2024). The argument combines two recent results: an Alexandrov estimate of Payne-Redaelli (2025) for locally semiconvex functions based on the area formula and a determinant majorization estimate of Harvey-Lawson (2024). The determinant majorization estimate has as a special case the arithmetic - geometric mean inequality, so the result includes the classical Alexandrov-Bakelman-Pucci estimate for linear operators. A potential theoretic approach is used involving subequation subharmonics and their dual subharmonics. Semiconvex approximation plays a crucial role.

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F. Reese Harvey, Kevin R. Payne. 2025-09-13. Alexandrov estimates for polynomial operators by determinant majorization. https://arxiv.org/abs/2509.10879

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