# Approximation in a Sobolev Space

Consider the open unit disk $$\mathbb{D}$$ in $$\mathbb{R}^2$$. In my analysis course, we defined the Sobolev space $$H^1(\mathbb{D})$$ in a somewhat unusual way. More precisely, $$H^1(\mathbb{D})$$ was defined to be the completion of the space $$\left\{ f \in C^1(\mathbb{D}) : \left\Vert{f}\right\Vert_{H^1(\mathbb{D})} < \infty \right\}$$ where $$\left\Vert f \right\Vert_{H^1(\mathbb{D})} = \left( \int_\mathbb{D} \left\vert f \right\vert^2 + \left\vert \nabla f \right\vert^2 \right)^{1/2}.$$ Now consider the function $$f(x) = \ln\left( \ln\left(1 + \frac{1}{\left\vert x \right\vert}\right)\right)$$ on $$\mathbb{D}$$. According to the usual weak derivative'' definition of Sobolev spaces (i.e. that used in Evans), I can prove that $$f \in H^1(\mathbb{D})$$. However, I am unsure of to establish this with the definition we are using.

My attempts so far have involved seeking out functions in $$C^1(\mathbb{D})$$ converging pointwise a.e. to $$f(x)$$ and checking whether this sequence also converges to $$f$$ with respect to the $$H^1$$-norm. For instance, I considered the sequence in $$C^1(\mathbb{D})$$ given by $$f_n(x)= \ln\left( \ln\left(1 + \frac{1}{\sqrt{x_1^2 + x_2^2 + \frac{1}{n}}}\right)\right).$$ Here, I am writing $$x = (x_1,x_2) \in \mathbb{D}$$.

Unfortunately, I was unable to show that the sequence $$(f_n)$$ was even Cauchy in $$H^1(\mathbb{D})$$. Is this the right approach? Or would a different sequence work better? I think mollifiers may work but I was hoping to avoid this in favour of an explicit sequence.

• Are you defining $|x|^2:=x_1^2 +x_2^2$ ? – irchans Oct 7 '18 at 2:50
• Yes, I am using the notation $x = (x_1,x_2) \in \mathbb{D}$. Thank you for pointing this out, I will add this to the question. – rolandcyp Oct 7 '18 at 2:52

$$f(x)=\log\left(\log\left(1+1/|x|\right)\right).$$
$$\mathrm{grad}f(x)= \frac{-x}{|x|^3\left(1+1/|x|\right)\log\left(1+1/|x|\right)}.$$
Let $$z_n = 1/(e^n - 1)$$, then $$f(z_n, 0) =\log(n)$$ and $$\mathrm{grad}f(z_n, 0)=(-\frac{(\exp(n/2)-\exp(-n/2))^2}n,0).$$
Define $$f_n(x) := f(x)$$ when $$|x|\geq z_n$$. When $$|x| let $$f_n$$ be the paraboloid which matches the gradient of $$f$$ and value of $$f$$ on the circle where $$|x|=z_n$$ with vertex at the origin. The $$f_n$$ are $$C^1$$ and they converge to $$f(x)$$ in the $$H^1$$norm.