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A property for the Monge-Ampère equation
Israel Journal of Mathematics ( IF 0.8 ) Pub Date : 2020-03-01 , DOI: 10.1007/s11856-020-1997-9
Daniele Puglisi

Let Ω ⊆ ℝ n be a non-empty open bounded set and h : Ω → ℝ be a non-negative continuous function. We prove that for any u ∈ C 2 (Ω) ∩ C 1 ( $$\overline{\Omega}$$ ) solution of the Monge–Ampère equation $${\rm{det}}(D^2u)=h\;\;{\rm{in}}\;\Omega,$$ , then ∇ u satisfies the convex hull property: ∇ u (Ω) ⊆ conv(∇ u ( ∂ Ω)).

中文翻译:

Monge-Ampère 方程的性质

设Ω ⊆ ℝ n 是一个非空开有界集,h : Ω → ℝ 是一个非负连续函数。我们证明对于任何 u ∈ C 2 (Ω) ∩ C 1 ( $$\overline{\Omega}$$ ) Monge–Ampère 方程 $${\rm{det}}(D^2u)=h \;\;{\rm{in}}\;\Omega,$$ ,那么 ∇ u 满足凸包性质:∇ u (Ω) ⊆ conv(∇ u (∂ Ω))。
更新日期:2020-03-01
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