  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.
