Partial Legendre transforms of non-linear equations

Details Partial Legendre transforms of non-linear equations Partial Legendre transforms of non-linear equations

2010-10-11

arxiv.org/abs/1010.2111v1

Mathematics

Analysis of PDEs

12 pages, no figure

Scientific paper

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.

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