Solutions of some Monge–Ampère equations with isolated and line singularities
Solutions of some Monge–Ampère equations with isolated and line singularities
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DOI：
10.1016/j.aim.2015.11.029
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发表时间：
2012-12
影响因子：
1.7
通讯作者：
Tianling Jin;Jingang Xiong
中科院分区：
文献类型：
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作者：
Tianling Jin;Jingang Xiong
本文研究了一类具有孤立奇点和线奇点的Monge-Ampère方程解的存在性、正则性、分类和渐近行为。我们对Rn中具有一个穿刺点的det∇2u=1的所有解进行了分类。这可以用来刻画椭球体，就像Serrin对拉普拉斯算子的超定问题一样。在k>；1有k个不可除奇点，模仿射等价的情况下，所有广义解的集合可以被标识为一个显式奥分叉。我们还建立了广义解的整体解的存在性和正则性，以及广义解在由点或直线组成的奇点附近的二阶导数的最优估计。几何动因来自奇异半平坦的Calabi-Yau度规。
In this paper, we study existence, regularity, classification, and asymptotic behaviors of solutions of some Monge–Ampère equations with isolated and line singularities. We classify all solutions of det∇ 2 u= 1 in R n with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having k non-removable singular points for k> 1, modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi–Yau metrics.