[论文解读] Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

Francescopaolo Montefalcone|ArXiv.org|Oct 29, 2009

Geometric Analysis and Curvature Flows参考文献 51被引用 3

一句话总结

本论文通过发展水平Coarea公式、在特征点处成立的爆破定理以及水平perimeter的一般化一阶变分公式，在子黎曼Carnot群中的光滑超曲面上建立了等周不等式、Sobolev不等式和Poincaré不等式。关键贡献是引入了一条涉及水平平均曲率的精确等周不等式，该不等式蕴含了在具有受控特征集的紧致超曲面上的相关Sobolev与Poincaré型不等式。

ABSTRACT

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [63] and, independently by Allard [1]. In particular, from this result one may deduce some related Sobolev-type inequalities; see Section 7. The strategy of the proof is inspired by the classical one. In particular, we shall begin by proving some linear isoperimetric inequalities. Once this is proven, one can deduce a local monotonicity formula and then conclude the proof by a covering argument. We stress however that there are many differences, due to our different geometric setting. Some of the tools which have been developed ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for characteristic points, and a smooth Coarea Formula for the HS-gradient; see Section 3 and Section 4. Other tools are the horizontal integration by parts formula and the 1st variation of the H-perimeter already developed in [68], [69], and here generalized to hypersurfaces having non-empty characteristic set. Some natural applications of these results are in the study of minimal and constant horizontal mean curvature hypersurfaces. Moreover we shall prove some purely horizontal, local and global Poincaré-type inequalities as well as some related facts and consequences; see Section 4 and Section 5.

研究动机与目标

• 将经典的等周不等式、Sobolev不等式与Poincaré不等式推广至Carnot群的子黎曼几何设定中。
• 为具有或不具有边界的C^2-光滑紧致超曲面（包括具有非空特征集的超曲面）建立这些不等式。
• 发展诸如H-梯度的水平Coarea公式以及在特征点处成立的爆破定理等几何工具。
• 将H-perimeter的一阶变分公式推广至具有非零特征集的超曲面，从而实现对奇点处的分析。
• 从该几何设定中推导出全局与局部Poincaré型不等式，以及相关的Sobolev嵌入。

提出的方法

• 通过证明线性等周不等式，借鉴Michael与Simon以及Allard的经典证明策略。
• 引入H-梯度的水平Coarea公式，实现对水平距离函数的水平集上的积分。
• 发展一种在特征点处也成立的H-perimeter爆破定理，这对局部分析至关重要。
• 利用由线性等周不等式导出的弱单调性公式与覆盖论证，证明主要的等周不等式。
• 应用强线性不等式及其单调性公式，推导H-perimeter的渐近行为与全局不等式。
• 将H-perimeter的一阶变分公式推广至具有非空特征集的超曲面，从而实现对奇异点的分析。

实验结果

研究问题

• RQ1 能否在k步Carnot群中为C^2-光滑超曲面建立一条精确的等周不等式，使其包含水平平均曲率？
• RQ2 经典证明策略（Michael与Simon）如何适应于具有非可积分布的子黎曼设定？
• RQ3 特征点在子黎曼超曲面上的等周与Sobolev理论中起何种作用？
• RQ4 能否从该几何设定中的等周不等式推导出全局与局部Poincaré型不等式？
• RQ5 Carnot群中超曲面上的精确Sobolev嵌入是什么？它们如何依赖于齐次维数Q？

主要发现

• 证明了在k步Carnot群中C^2-光滑紧致超曲面S上存在等周不等式，其涉及水平平均曲率与齐次(Q-1)-维Hausdorff测度。
• 建立了全局Sobolev型不等式：对任意ψ ∈ C^1_0(S)，ψ的L^{(Q-1)/(Q-2)}-范数被一个常数乘以|ψ|的L^1-范数与|grad_H S ψ|的L^1-范数之和所控制。
• 当p ∈ [1, Q-1[ 时，存在Sobolev嵌入：对所有q ∈ [p, p*]，ψ的L^q(S)-范数被一个常数乘以ψ的L^p(S)-范数与|grad_H S ψ|的L^p(S)-范数之和所控制。
• 当p = Q-1时，ψ的L^q(S)-范数对所有q ∈ [Q-1, ∞[ 有界，且对范数的控制方式相同。
• 推导出一条强线性等周不等式，其蕴含H-perimeter的单调性公式，该公式被用于分析H-perimeter在点附近的渐近行为。
• 论文证明了在假设(H)或(H2)下，相对紧开集U ⊂ S的边界H-perimeter被H-梯度距离函数的总变差的常数倍所控制。