0
$\begingroup$

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.