Geometric properties of the sections of solutions to the Monge-Ampère equation
Geometric properties of the sections of solutions to the Monge-Ampère equation
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DOI：
10.1090/s0002-9947-00-02491-0
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发表时间：
2000-05
影响因子：
1.3
通讯作者：
C. E. Gutiérrez;Qingbo Huang
中科院分区：
文献类型：
--
作者：
C. E. Gutiérrez;Qingbo Huang
本文建立了Monge-Ampere方程det D2 =的广义解4的截面的几个几何性质，当测度t满足加倍性质时.一个主要的结果是一个特征的加倍措施，t在4的横截面的几何性质。这用于获得在适当的归一化下有效的横截面的形状和不变性的估计。0.设：RI -+ R是凸函数。在点（xo，q（xo））处q的支撑超平面是仿射函数？(x)= q（xo）+p.（x xo）使得q（x）>？(x)对于所有x e R '。给定t > 0，在高度t处的0的部分是凸集So（xo，p，t）= {x ∈ R：+（x）'P（RI），其由Vq$（xo）= {p：q（x）>？(xo)+p（x-x0），Vx ∈ Rn}。如果E C Rn，则V+（E）= UXEE Vq（x）。根据亚历山德罗夫的一个经典定理，使得V+（E）是勒贝格可测的集合E的类是一个包含Borel集的（o-代数，并且可以定义与q相关的Monge-Ampere测度作为Borel测度，u由pt V+（= E）给出;参见[Ch-Y]。本文详细分析了凸函数q在其Monge-Ampere测度u满足加倍条件时的截面的几个重要几何性质。对这些性质的兴趣来自于对Monge-Ampere方程及其线性化（椭圆和抛物）的研究，以及真实的调和分析;见[Cl]、[C2]、[C-G1]、[C-G2]和[H]。其中一些属性已被用于在这些参考文献中证明一个引理的贝西科维奇的类型和卡尔德龙-齐格蒙特分解ir条款的部分，使我们能够建立估计的解决方案的线性化蒙赫-安培方程。此外，其中一些性质意味着R'由编辑于1997年6月9日收到。1991年数学学科分类。小学35 J60、35 D10;中学26 B25。
In this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the Monge-Ampere equation det D2 = , when the measure ,t satisfies a doubling property. A main result is a characterization of the doubling measures ,t in terms of a geometric property of the cross sections of 4. This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations. 0. INTRODUCTION Let : RI -+ R be a convex function. A supporting hyperplane to q at the point (xo, q(xo)) is an affine function ?(x) = q(xo) + p. (x xo) such that q(x) > ?(x) for all x e R'. Given t > 0, a section of 0 at height t is the convex set So (xo,p,t) = {x e R : +(x) 'P(RI) defined by Vq$(xo) = {p: q(x) > ?(xo) +p (x-xo), Vx E Rn}. If E C Rn, then V+(E) = UXEE Vq(x). By a classical theorem of Aleksandrov, the class of sets E such that V+(E) is Lebesgue measurable is a (o-algebra that contains the Borel sets and one can define the Monge-Ampere measure associated with q as the Borel measure ,u given by pt V+( = E); see [Ch-Y]. The purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated Monge-Ampere measure ,u satisfies a doubling condition. The interest in these properties comes from the study of the solutions of the Monge-Ampere equation, and its linearizations, both elliptic and parabolic, and from real harmonic analysis; see [Cl], [C2], [C-G1], [C-G2] and [H]. Some of these properties have been used in those references to prove a lemma of Besicovitch's type and a Calderon-Zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized Monge-Ampere equation. Also, some of these properties imply that R' Received by the editors June 9, 1997. 1991 Mathematics Subject Classification. Primary 35J60, 35D10; Secondary 26B25.