A maximum rank problem for degenerate elliptic fully nonlinear equations

TL;DR

This paper proves that solutions to certain degenerate elliptic fully nonlinear equations, including the Monge-Ampère and Donaldson equations, have maximum rank properties in low dimensions, revealing structural insights into their solutions.

Contribution

It establishes maximum and constant rank properties for solutions of the Monge-Ampère and Donaldson equations in low dimensions, advancing understanding of their solution structure.

Findings

Solutions have maximum rank when n ≤ 2

Constant rank property for Donaldson equation when n=3

Structural insights into degenerate elliptic equations

Abstract

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in $n+1$ dimensions, namely the real Monge-Amp\`ere equation and the Donaldson equation, are shown to have maximum rank in the space variables when $n \leq 2$. A constant rank property is also established for the Donaldson equation when $n=3$.