Properties of the solutions of the linearized Monge-Ampère equation
Properties of the solutions of the linearized Monge-Ampère equation
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DOI：
10.1353/ajm.1997.0010
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发表时间：
1997-04
影响因子：
1.7
通讯作者：
L. Caffarelli;C. E. Gutiérrez
中科院分区：
文献类型：
--
作者：
L. Caffarelli;C. E. Gutiérrez
设Φ： R n→R为严格凸光滑函数，μ = det d2 Φ为Φ生成的蒙日-安培量。给定x∈rn， t为0，设S (x, t) = {y∈rn： Φ(y) x) +∇Φ(x)·(y - x) + t}。本文的目的是研究由a ij (x) D ij u = 0给出的线性化蒙日-安培方程解的性质，其中系数a ij (x)是矩阵d2 Φ(x)的协因子。假设μ在集合S （x, t）上满足倍性条件，并在每一尺度上满足勒贝格测度的一致连续性条件。我们建立了非负解u在高度t处的分布函数衰减为t的负次幂，并证明了在截面S （x, t）上的一个不变的哈纳克不等式。所有的估计都不依赖于Φ的规律性，而只依赖于对μ测度所作假设中的常数。
Let Φ: R n → R be a function strictly convex and smooth, and μ = det D 2 Φ is the Monge-Ampere generated by Φ. Given x ∈ R n and t 0, let' S ( x, t ) = { y ∈ R n : Φ( y ) x ) + ∇Φ( x ) · ( y - x ) + t }. The purpose of this paper is to study the properties of the solutions of the linearized Monge-Ampere equation given by a ij ( x ) D ij u = 0 where the coefficients a ij ( x ) are the cofactors of the matrix D 2 Φ( x ). It is assumed that μ satisfies a doubling condition on the sets S ( x, t ) and a uniform continuity condition at every scale with respect to Lebesgue measure. We establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant Harnack's inequality on the sections S ( x, t ). All the estimates are independent of the regularity of Φ and depend only on the constants in the hypotheses made on the measure μ.