A maximum rank problem for degenerate elliptic fully nonlinear equations

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in $n+1$ dimensions, namely the real Monge-Ampère 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 $n=3$.

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