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arXiv · 2312.03280

On global $W^{2,\delta}$ estimates for the Monge-Amp\`ere equation on general bounded convex domains

Nam Q. Le
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Abstract

We establish global $W^{2,\delta}$ estimates, for all $\delta<\frac{1}{n-1}$, for convex solutions to the Monge-Amp\`ere equation with positive $C^{2,\beta}$ right-hand side and zero boundary values on general bounded convex domains in ${\mathbb R}^n$ ($n\geq 2$). We exhibit examples showing that global $W^{2, \frac{n}{2(n-1)}}$ estimates fail in all dimensions, so the range of $\delta$ is sharp in two dimensions.

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Nam Q. Le. 2023-12-06. On global $W^{2,\delta}$ estimates for the Monge-Amp\`ere equation on general bounded convex domains. https://arxiv.org/abs/2312.03280

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