My question is the regular definition of the space $\mathcal{D}^{1,2}(\Omega)$, for some open $\Omega \subset \mathbb{R}^3$.
I know this space is usefull for the studies of solution for PDE problems. I jus't found some equivalences around the books.
Sorry for the weak question, but I can't found this in bibliography. Every help will be appreciated!
For an open set $\Omega \subseteq \mathbb{R}^n$, $m \in \mathbb{N}$, and $1 \le p < \infty$ we often define $\mathcal{D}^{m,p}(\Omega)$ for the space obtained by taking the completion of $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ relative to the norm $$ \Vert u \Vert_{\mathcal{D}^{m,p}} = \left(\sum_{|\alpha| = m} \Vert \partial^\alpha u \Vert_{L^p}^p \right)^{1/p}. $$ Note crucially that we only look at the $m^{th}$ order derivatives, but this does define a norm on $\mathcal{D}(\Omega)$ due to the compact support condition. The $\mathcal{D}$ part of the notation comes from the connection to distributions, where $\mathcal{D}$ and $\mathcal{D}'$ are used frequently.