Partial Regularity for Singular Solutions to the Monge‐Ampère Equation
Partial Regularity for Singular Solutions to the Monge‐Ampère Equation
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Monge-Ampère 方程奇异解的部分正则性
DOI：
10.1002/cpa.21534
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发表时间：
2013
影响因子：
3
通讯作者：
Connor Mooney
中科院分区：
文献类型：
--
作者：
Connor Mooney
证明了在一个奇异的Hausdorff （n‐1）维测度0的集合上，Monge‐ampontre不等式detD2u≥1的解是严格凸的。此外，我们通过构造det D2u = 1的解来证明这是最优的，该解具有尽可能接近于n‐1的豪斯多夫维数奇异集。由此，我们得到了右侧有界的Monge‐ampontre方程的w2,1正则性和右侧有充分正则的Monge‐ampontre方程的唯一连续性。©2015 Wiley期刊公司
We prove that solutions to the Monge‐Ampère inequality detD2u≥1 in ℝn are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D2u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.