Let $\Omega = \{ x \in \mathbb{R} : | x | \leq 1 \}$. Show that the function $v(x) = |x|^{\alpha}$ belong to $H^1(\Omega)$ if $\alpha > 0$

Let $$\Omega = \{ x \in \mathbb{R} : | x | \leq 1 \}$$. Show that the function $$v(x) = |x|^{\alpha}$$ belong to $$H^1(\Omega)$$ if $$\alpha > 0$$.

We know that:

$$\mathbb{L}_2(\Omega) = \{ v: v \ \text{its defined in} \ \Omega \ \text{and} \ \int_{\Omega} v^2 dx < \alpha \}$$

and that

$$H^1(\Omega) = \{ v \in \mathbb{L}_2 (\Omega) : \frac{\partial v}{ \partial x_i} \in \mathbb{L}_2(\Omega) , i = 1 , 2 , \ldots , d \}$$

where $$d \text{ comes from } R^d$$

I did'nt understand how to solve this problem, i have tried numerically but my teacher says that im going in the wrong direction

Appreciate any help

thanks

• First question: do you know that $v \in L^2(\Omega)$? – Umberto P. Feb 20 at 18:39
• yes, what you can deduce from that ? – Saiten Feb 20 at 19:10
• i've thought that i would have to prove that when $\alpha \leq 0$ it does not belong to $\mathbb{L}_2 (\Omega)$ . – Saiten Feb 20 at 19:14
• Good. Second question: what is $\dfrac{\partial v}{\partial x_i}$? – Umberto P. Feb 20 at 21:53
• Away from the origin $v(x)$ is classically differentiable. You need to be able to calculate $\dfrac{\partial v}{\partial x_i}$. – Umberto P. Feb 21 at 1:27