A maximum rank problem for degenerate elliptic fully nonlinear equations (Q453324)

In this paper, the authors study a maximum rank problem to degenerate elliptic Dirichlet problems for the two basic models defined on the space \((\mathbb R/\mathbb Z)^n\times (0,1)\): The Monge-Ampère equation and the Donaldson equation whose both right hands are equal to \(\epsilon\), \(0<\epsilon <<1\). They prove that, for \(n\leq 2\), the solutions of the two models have the maximum rank when the boundary data satisfy the maximum rank property at first. A constant rank property is also established for the Donaldson equation when \(n=3\).