高维球具有“外多内少”的特点

hbghlyj

Examples
In higher dimensions, it is no longer true that, for example, \(W^{1,1}\) contains only continuous functions. For example, \(|x|^{-1} \in W^{1,1}(\mathbb{B}^3)\) where \(\mathbb{B}^3\) is the unit ball in three dimensions. For \(k > n/p\), the space \(W^{k,p}(\Omega)\) will contain only continuous functions, but for which \(k\) this is already true depends both on \(p\) and on the dimension. For example, as can be easily checked using spherical polar coordinates for the function \(f : \mathbb{B}^n \to \Bbb R \cup \{\infty \}\) defined on the $n$-dimensional ball we have:
\[f(x) = | x |^{-\alpha} \in W^{k,p}(\mathbb{B}^n) \Longleftrightarrow \alpha < \tfrac{n}{p} - k.\]
Intuitively, the blow-up of $f$ at 0 counts for less when $n$ is large since the unit ball has more outside and less inside in higher dimensions.

如何理解？

hbghlyj

用极坐标计算，\begin{aligned}
\int_{B_1(0)}|x|^{-\alpha p} \mathrm{~d} x & =\int_0^1 \int_{\partial B_r(0)}|x|^{-\alpha p} \mathrm{~d} S_x \mathrm{~d} r \\
& =r^{n-1}\omega_{n-1} \int_0^1 r^{-\alpha p} \mathrm{~d} r\\
& =\omega_{n-1} \int_0^1 r^{-\alpha p+n-1} \mathrm{~d} r<\infty⇔n − αp > 0
\end{aligned}其中$\omega_{n-1}=\int_{\partial B_1(0)}\mathrm{d}S_x$.

hbghlyj

设$u(x)=\abs{x}^\alpha$.
设$\alpha<\frac{n}{p}$，根据2#，$u\in\mathrm L^p_\text{loc}$
Claim: $u \in \mathrm{W}^{1, p}\left(B_1(0)\right)$ precisely for $\alpha<\frac{n}{p}-1$ when $p \in[1, n)$ (and never when $\alpha>0$ and $p \geqslant n$).
计算偏导：$\partial_j u=-\alpha|x|^{-\alpha-2} x_j$ 对每个 $1 \leqslant j \leqslant n$.
若$u∈\mathrm{W}^{1, p}$，则对每个$j$都有$\partial_j u \in \mathrm{L}^p(𝔹)$，这等价于它们的平方平均$\in \mathrm{L}^p(𝔹)$：
\[
\left(\sum_{j=1}^n\left|\partial_j u\right|^2\right)^{\frac{1}{2}}=\alpha|x|^{-\alpha-1} \in \mathrm{L}^p(𝔹)
\]
根据2#（把2#的$\alpha$换成$\alpha+1$）得，$n-(\alpha+1) p>0$.

hbghlyj

hbghlyj 发表于 2024-3-22 18:16
For \(k > n/p\), the space \(W^{k,p}(\Omega)\) will contain only continuous functions

如何理解？和Sobolev Embedding Theorem有关？