Partial Legendre transforms of non-linear equations

Pengfei Guan; D. H. Phong

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arXiv · 1010.2111

Partial Legendre transforms of non-linear equations

Pengfei Guan·D. H. Phong

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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.

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Pengfei Guan, D. H. Phong. 2010-10-11. Partial Legendre transforms of non-linear equations. https://arxiv.org/abs/1010.2111

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