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

Campanato’s criterion is a cornerstone in the regularity theory for variational problems, as it characterizes Hölder continuity in terms of the uniform decay of certain integral averages. We shall use this criterion in Section 26.2.

Theorem 6.1 (Campanato’s criterion) If $n \geq 1, p \in[1, \infty), \gamma \in(0,1]$, then there exists a constant $C(n, p, \gamma)$ with the following property. If $u \in L^p(B)$,
$$(u){x, r}=\frac{1}{|B \cap B(x, r)|} \int{B \cap B(x, r)} u, \quad x \in B, r>0,$$
and there exists a constant $\kappa$ such that the uniform decay condition
$$\left(\frac{1}{r^n} \int_{B \cap B(x, r)}\left|u-(u){x, r}\right|^p\right)^{1 / p} \leq \kappa r^\gamma, \quad \forall x \in B,$$ holds true, then there exists a function $\bar{u}: B \rightarrow \mathbb{R}$ with $\bar{u}=u$ a.e. on $B$ and $$|\bar{u}(x)-\bar{u}(y)| \leq \kappa^{\prime}|x-y|^\gamma, \quad \forall x, y \in B .$$ where $\kappa^{\prime}=C(n, p, \gamma) \kappa$. Remark 6.2 It is easily seen that there exists a constant $c(n)>0$ such that $$c(n) r^n \leq|B \cap B(x, r)| \leq \omega_n r^n,$$ for every $x \in B, r>0$. In particular, if $u \in L^P(B)$, then by Theorem $5.16$ $$\lim {r \rightarrow 0^{+}} \frac{1}{r^n} \int_{B \cap B(x, r)}\left|u-(u)_{x, r}\right|^p=0,$$
for a.e. $x \in B$ (precisely, for every Lebesgue point $x$ of $1_R u \in L^1\left(\mathbb{R}^n\right)$ ).

## 数学代写|几何测度论代写geometric measure theory代考|Lower dimensional densities of a Radon measure

Given a Radon measure $\mu$ on $\mathbb{R}^n$ and $s \in(0, n]$, we define the upper $s$ dimensional density $\theta_s^(\mu): \mathbb{R}^n \rightarrow[0, \infty]$ of $\mu$ as $$\theta_s^(\mu)(x)=\limsup {r \rightarrow 0^{+}} \frac{\mu(\bar{B}(x, r))}{\omega_s r^s}, \quad x \in \mathbb{R}^n .$$ We note that, by Exercise $4.27, \theta_s^(\mu)$ is a Borel function. If $x \in \mathbb{R}^n$ is such that the limit in (6.7) exists, then we denote by $\theta_s(\mu)(x)$ this value, and call it the $s$-dimensional density of $\mu$ at $x$. If $\theta_s(\mu)(x)$ is defined at $x$, then closed balls may be replaced by open balls, that is, $$\theta_s(\mu)(x)=\lim {r \rightarrow 0^{+}} \frac{\mu(B(x, r))}{\omega_s r^s} ;$$
see Remark 5.7. Since $\omega_n r^n=|B(x, r)|$, looking at $n$-dimensional densities is equivalent to differentiating with respect to $\mathcal{L}^n$. Hence, the study of $n$-dimensional densities is fully addressed by the Lebesgue-Besicovitch differentiation theorem. The behavior of $s$-dimensional densities, when $s \in(0, n)$, is more complex. The following theorem and its corollary (which extend the identity $\theta_n(E)=0$ a.e. on $\mathbb{R}^n \backslash E$ to arbitrary values of $s$ ) illustrate what can be concluded in full generality, and will be used in Chapters 11,16 , and 17.
Theorem $6.4$ (Upper $s$-dimensional densities and comparison with $\mathcal{H}^s$ ) If $\mu$ is a Radon measure on $\mathbb{R}^n, M$ is a Borel set, and $s \in(0, n)$, then
$1 \leq \theta_s^(\mu) \quad$ on $M \quad \Rightarrow \quad \mathcal{H}^s(M) \leq \mu(M)$,
$\theta_s^(\mu) \leq 1 \quad$ on $M \quad \Rightarrow \quad \mu(M) \leq 2^s \mathcal{H}^s(M)$. Proof Step one: We prove (6.8). We may directly assume that $M \subset B_R$, for some $R>0$. We first prove that $\theta_s^(\mu) \geq 1$ on $M$ implies $\mathcal{H}^s(M)<\infty$. Given $\delta>0$, let us consider a family of closed balls
$$\mathcal{F}=\left{\bar{B}(x, r): x \in M, 2 r<\delta, \mu(\bar{B}(x, r)) \geq(1-\delta) \omega_s r^s\right} .$$

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

Campanato 准则是变分问题正则性理论的基石，因为它根据某些积分平均值的均匀亳减来表征 Hölder 连 续性。我们将在第 $26.2$ 节中使用这个标准。

$$(u) x, r=\frac{1}{|B \cap B(x, r)|} \int B \cap B(x, r) u, \quad x \in B, r>0,$$

$$\left(\frac{1}{r^n} \int_{B \cap B(x, r)}|u-(u) x, r|^p\right)^{1 / p} \leq \kappa r^\gamma, \quad \forall x \in B,$$

$$|\bar{u}(x)-\bar{u}(y)| \leq \kappa^{\prime}|x-y|^\gamma, \quad \forall x, y \in B .$$

$$c(n) r^n \leq|B \cap B(x, r)| \leq \omega_n r^n,$$

$$\lim {r \rightarrow 0^{+}} \frac{1}{r^n} \int{B \cap B(x, r)}\left|u-(u)_{x, r}\right|^p=0,$$

## 数学代写|几何测度论代写geometric measure theory代考|Lower dimensional densities of a Radon measure

$$\left.\theta_s^{(} \mu\right)(x)=\limsup r \rightarrow 0^{+} \frac{\mu(\bar{B}(x, r))}{\omega_s r^s}, \quad x \in \mathbb{R}^n .$$

$$\theta_s(\mu)(x)=\lim r \rightarrow 0^{+} \frac{\mu(B(x, r))}{\omega_s r^s} ;$$

$\left.\theta_s^{\prime} \mu\right) \leq 1$ 上 $M \Rightarrow \mu(M) \leq 2^s \mathcal{H}^s(M)$. 证明第一步：我们证明 (6.8)。我们可以直接假设
$M \subset B_R$ ，对于一些 $R>0$. 我们首先证明 $\left.\theta_s^{\prime} \mu\right) \geq 1$ 上 $M$ 暗示 $\mathcal{H}^s(M)<\infty$. 给定 $\delta>0$ ，让我们考虑一

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