Alexandrov estimates for polynomial operators by determinant majorization

2025/09/13 by Harvey, F. Reese, Payne, Kevin R. · 1 citation
#26B25 #35B45 #35B50 #35D40 #35J70 #35J96 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.10879

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.

Citations

• The Correspondence Principle: A bridge between general potential theories and nonlinear elliptic differential operators
• A primer on semiconvex functions in general potential theories
• Uniform entropy and energy bounds for fully non-linear equations
• Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive
• Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs
• Interior estimates for p-plurisubharmonic functions
• Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient
• Principal eigenvalues for k-Hessian operators by maximum principle methods
• The inhomogeneous Dirichlet Problem for natural operators on manifolds
• On viscosity solutions to the Dirichlet problem for elliptic branches of nonhomogeneous fully nonlinear equations
• Tangents to subsolutions -- existence and uniqueness, Part I
• Hyperbolic polynomials and the Dirichlet problem

Cited by

• The Correspondence Principle: A bridge between general potential theories and nonlinear elliptic differential operators

Related

• Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations
• Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method
• Uniform estimates: from Yau to Kolodziej
• New maximum principles for linear elliptic equations
• Uniform estimates for elliptic equations with Carathéodory nonlinearities at the interior and on the boundary