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arXiv · 1806.01720

Interior C^{1,1} regularity of solutions to degenerate Monge-Amp\`{e}re type equations

Feida Jiang·Juhua Shi·Xiaoping Yang
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Abstract

In this paper, we study the interior C^{1,1} regularity of viscosity solutions for a degenerate Monge-Amp\`{e}re type equation \det[D^{2}u-A(x, u, Du)]=B(x, u, Du) when B \geq 0 and B^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}\times\mathbb{R}\times \mathbb{R}^n). We prove that u\in C^{1,1}(\Omega) under the A3 condition and A3w^+ condition respectively. In the former case, we construct a suitable auxiliary function to obtain uniform {\it a priori} estimates directly. In the latter case, the main argument is to establish the Pogorelov type estimates, which are interesting independently.

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Feida Jiang, Juhua Shi, Xiaoping Yang. 2018-06-05. Interior C^{1,1} regularity of solutions to degenerate Monge-Amp\`{e}re type equations. https://arxiv.org/abs/1806.01720

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