# 数学代写|几何测度论代写geometric measure theory代考|MATH823

## 数学代写|几何测度论代写geometric measure theory代考|Lipschitz functions

The notion of Lipschitz function plays a special role in Geometric Measure Theory. Various metric properties which are characteristic of $C^1$-functions are also satisfied by Lipschitz functions. At the same time, the Lipschitz condition is stable under plain pointwise convergence and can be formulated in terms of set inclusions only, two features that make it particularly compatible with measure-theoretic arguments. In this chapter (and in the following) we address some fundamental properties of Lipschitz functions. In Section 7.1, we prove Kirszbraun’s theorem, which guarantees the existence of Lipschitz-constant preserving extensions. In Sections $7.2$ and $7.3$ Lipschitz functions are shown to possess bounded distributional gradients and to be a.e. classically differentiable (Rademacher’s theorem). Recall that, if $E \subset \mathbb{R}^n$ and $f: E \subset \mathbb{R}^n \rightarrow \mathbb{R}^m$, then is a Lipschitz function on $E$, provided
$$\operatorname{Lip}(f ; E)=\sup \left{\frac{|f(x)-f(y)|}{|x-y|}: x, y \in E, x \neq y\right}<\infty .$$
We simply set $\operatorname{Lip}(f)=\operatorname{Lip}\left(f ; \mathbb{R}^n\right)$. The geometric nature of the Lipschitz condition is suggested by the following remark.
Remark 7.1 If $\Gamma\left(f ; \mathbb{R}^n\right)$ denotes the graph of $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ over $\mathbb{R}^n$, that is
$$\Gamma\left(f ; \mathbb{R}^n\right)=\left{(y, f(y)) \in \mathbb{R}^n \times \mathbb{R}^m: y \in \mathbb{R}^n\right},$$
then $f$ is a Lipschitz function on $\mathbb{R}^n$, provided that, for every $x \in \mathbb{R}^n$, the graph of $f$ is contained in the “cone” of vertex $(x, f(x))$ and “opening” $\operatorname{Lip}(f)$, that is (see Figure 7.1),
$$\Gamma\left(f ; \mathbb{R}^n\right) \subset \bigcap_{x \in \mathbb{R}^n}(x, f(x))+\left{(z, w) \in \mathbb{R}^n \times \mathbb{R}^m:|w| \leq \operatorname{Lip}(f)|z|\right}$$

## 数学代写|几何测度论代写geometric measure theory代考|Kirszbraun’s theorem

Theorem $7.2$ (Kirszbraun’s theorem) If $E \subset \mathbb{R}^n$ and $f: E \rightarrow \mathbb{R}^m$ is a Lipschitz function, then there exists $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $g=f$ on $E$ and $\operatorname{Lip}(g)=\operatorname{Lip}(f ; E)$.
When $m=1$, a (somewhat) explicit extension $g$ of $f$ is defined as
$$g(y)=\inf {f(x)+\operatorname{Lip}(f)|y-x|: x \in E}, \quad y \in \mathbb{R}^n .$$
Having in mind Remark 7.1, (7.1) defines the maximal extension of $f$.
Lemma 7.3 (McShane’s lemma) If $E \subset \mathbb{R}^n$ and $f: E \rightarrow \mathbb{R}$ is a Lipschitz function on $E$, then the function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ defined in (7.1) satisfies $g=f$ on $E$ and $\operatorname{Lip}(g)=\operatorname{Lip}(f ; E)$.

Proof Clearly, $g \leq f$ on $E$. Since $f(x)+\operatorname{Lip}(f)|y-x| \geq f(y)$ for every $x, y \in E$, minimizing over $x \in E$ we find $g \geq f$ on $E$. Now, if $x, y, z \in \mathbb{R}^n$, then $g(y) \leq f(x)+\operatorname{Lip}(f ; E)|y-x| \leq(f(x)+\operatorname{Lip}(f ; E)|z-x|)+\operatorname{Lip}(f ; E)|z-y|$.
Minimizing over $x \in E$ we find $g(y) \leq g(z)+\operatorname{Lip}(f ; E)|z-y|$, and then, by symmetry, $|g(y)-g(z)| \leq \operatorname{Lip}(f ; E)|y-z|$.

When $m>1$ we can extend each component $f^{(i)}$ of $f$ by McShane’s lemma, thus finding a Lipschitz function $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with $g=f$ on $E$. This extension, however, will merely satisfy the non-optimal bound $\operatorname{Lip}(g) \leq \sqrt{m} \operatorname{Lip}(f ; E)$. We thus need a different strategy to prove Kirszbraun’s theorem. We shall use the following geometric lemma; see Figure 7.2.

Lemma 7.4 Given a finite collection of closed balls $\left{\bar{B}\left(x_k, r_k\right)\right}_{k=1}^N$ in $\mathbb{R}^n$, set
$$C_t=\bigcap_{k=1}^N \bar{B}\left(x_k, t r_k\right), \quad t \geq 0 .$$
If $s=\inf \left{t \geq 0: C_t \neq \emptyset\right}$, then $s<\infty$ and $C_s$ reduces to a single point $x_0$, which belongs to the convex hull of those $x_k$ such that $\left|x_0-x_k\right|=s r_k$.

## 数学代写|几何测度论代写geometric measure theory代考|Lipschitz functions

Lipschitz 函数的概念在几何测度论中起着特殊的作用。各种度量属性，这些属性是 $C^1$ – 函数也满足
Lipschitz 函数。同时，Lipschitz 条件在简单的逐点收玫下是稳定的，并且可以仅根据集合包含来表述，这 两个特征使其与测度论论证特别兼容。在本章（以及接下来的章节) 中，我们将讨论 Lipschitz 函数的一些 基本性质。在第 $7.1$ 节中，我们证明了 Kirszbraun 定理，它保证了 Lipschitz 常数保持扩展的存在。在部分 7.2和7.3Lipschitz 函数被证明具有有界分布梯度并且是经典可微的 (Rademacher 定理) 。回想一下，如 果 $E \subset \mathbb{R}^n$ 和 $f: E \subset \mathbb{R}^n \rightarrow \mathbb{R}^m$ ，那么是一个 Lipschitz 函数 $E$ ，假如

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## 数学代写|几何测度论代写geometric measure theory代考|Kirszbraun’s theorem

$$g(y)=\inf f(x)+\operatorname{Lip}(f)|y-x|: x \in E, \quad y \in \mathbb{R}^n .$$

$g(y) \leq f(x)+\operatorname{Lip}(f ; E)|y-x| \leq(f(x)+\operatorname{Lip}(f ; E)|z-x|)+\operatorname{Lip}(f ; E)|z-y|$.

$|g(y)-g(z)| \leq \operatorname{Lip}(f ; E)|y-z|$.

$$C_t=\bigcap_{k=1}^N \bar{B}\left(x_k, t r_k\right), \quad t \geq 0 .$$

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