I'm beginning with Sobolev spaces and I found out, that
$$ H^k = W^{k,2}. $$
I've also seen the following exercise recently:
$$ \frac{1}{2}u'' = 1 $$
And here I'm supposed to find out if $u$ belongs to $H^1, H^2$ etc.
Could you, please, explain it to me?
I know, that Sobolev space has a norm
$$ ||u||_{k,p} = \left( \sum_{|\alpha| \leq k} \int_\Omega \left| D^\alpha u(x) \right|^p dx \right)^{\frac{1}{p}} $$
and that it's a complete space. But still, I'm not sure about a way, how to show, that some function belongs into it.
You should know that for each $f \in L^{p}_{loc}(\Omega)$ we can define a Distribution by:
$T_{f}: \mathcal{D}(\Omega) \rightarrow\mathbb{R}$, where $T_{f}(\phi) = \displaystyle\int_{\Omega}f\phi$.
In the case, $T_{f}$ is called regular distribution.
The Du Bouis Reymond Lemma states that the map $f\mapsto T_{f}$ is one-to-one. However, there are Distributions which can't associated with $f \in L^{p}(\Omega)$ (Dirac's Distributions is a classic example).
Now consider the Heaviside's function, clearly this function belongs to $L^{1}_{loc}(\Omega)$, so can be associated with a distribution, but the derivative of this distribution can't be associated with a function in $L^{1}_{loc}(\Omega)$. So, a problem arises, because I would like (for some reason) that derivatives of regular distributions, was regular too. For this reason, the Sobolev spaces arises.
For exmaple, the space $W^{1,2}(\Omega) = H^{1}(\Omega) = \lbrace u\in L^{2}(\Omega); \dfrac{\partial u}{\partial x_{i}} \in L^{2}(\Omega)\rbrace$.
The derivate in this definition, is the derivative of distribution associated to $u$.