Proving that a certain function is in $W^{1,n}(B(0,1))$

Question

Fix $0<\alpha<1-\frac{1}{n}$ and let $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$ be the function $f(x)=(\log(\frac{1}{|x|}))^{\alpha}$.

How can I prove that $f\in W^{1,n}(B(0,1))$?

Answer 1

First $f = (-\ln|x|)^{\alpha}$, hence $$ \nabla f = -\alpha (-\ln|x|)^{\alpha - 1} \frac{x}{|x|^2} $$ and $$ |\nabla f | = \alpha (-\ln|x|)^{\alpha - 1} \frac{1}{|x|} $$ Let $r = |x|$, then $0<r<1$, change the integral over spheres: $$ \int_{B(0,1)} |f|^n \,dx = \int^1_0 \left(\int_{\partial B(0,r)} |f|^n dS\right)\,dr , $$ where $|f| = |\ln r|^{\alpha}$ is a constant on the surface of the $n$-sphere $\partial B(0,r)$, and we can pull it out from the surface integral: $$ \int_{B(0,1)} |f|^n \,dx = \int^1_0 \left(\int_{\partial B(0,r)} 1 dS\right)|\ln r|^{\alpha n} \,dr = c(n)\int^1_0 |\ln r|^{\alpha n} r^{n-1} dr , $$ where $\int_{\partial B(0,r)} 1 dS = c(n)r^{n-1}$ is the surface area of the $n$-sphere. This integral might behave singular when $r\to 0$. For $\alpha n< n-1$, and $$ r^{n-1}<|\ln r|^{\alpha n} r^{n-1} < |\ln r|^{n-1} r^{n-1} $$ when $0<r<1$. Now since $$ \int |\ln r|^{n-1} r^{n-1} \,dr\sim O(r^n |\ln r|^{n-1}), \quad \text{ and }\int r^{n-1} \,dr\sim O(r^n) $$ and $$ \lim_{r\to 0} r^n |\ln r|^{n-1} = 0 $$ we have $$ \int_{B(0,1)} |f|^n \,dx <\infty. \tag{1} $$

For $$ \begin{aligned} &\int_{B(0,1)} |\nabla f|^n \,dx = \int_{B(0,1)} \alpha (-\ln|x|)^{n\alpha - n} \frac{1}{|x|^n} \, dx \sim c(n) \int^1_0 \alpha |\ln r|^{n\alpha - n} \frac{1}{r^n} r^{n-1} \,dr \\ &= \int^1_0 \alpha |\ln r|^{n\alpha - n} \frac{1}{r}\,dr = \int^1_0 \alpha |\ln r|^{n\alpha - n}\,d(\ln r) \sim O(|\ln r|^{n\alpha - n + 1}) \end{aligned} $$ For $-n +1 <n\alpha - n + 1 < 0$, for $n\geq 2$ here (otherwise $\alpha$ will not exist), we have $$ \int_{B(0,1)} |\nabla f|^n \,dx \sim O(|\ln r|^{-\epsilon}) $$ for some $\epsilon >0$, hence $$ \int_{B(0,1)} |\nabla f|^n \,dx < \infty.\tag{2} $$ The claim follows from (1) and (2).

My argument is kinda rough for I used big-O notation to represent quantity in the same order ignoring the constant, I suggest you carry out the calculation more explicitly if this is a homework.