Partial Legendre transforms of non-linear equations

Abstract: The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Amp`ere equation is another equation of Monge-Amp`ere type. In 1+1 dimensions, this can be applied to obtain uniform estimates to all orders for the degenerate Monge-Amp`ere equation with boundary data satisfying a strict convexity condition.