Partial Legendre transforms of non-linear equations

TL;DR

This paper demonstrates that the partial Legendre transform of certain non-linear elliptic equations results in another elliptic equation, enabling new estimates for degenerate Monge-Ampère equations in specific dimensions.

Contribution

It establishes that the partial Legendre transform preserves ellipticity for a class of non-linear equations, including Monge-Ampère, and applies this to derive uniform estimates in 1+1 dimensions.

Findings

Partial Legendre transform of non-linear elliptic equations yields another elliptic equation.

Transform of Monge-Ampère equation remains of Monge-Ampère type.

Application to 1+1 dimensions provides uniform estimates for degenerate Monge-Ampère 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.