what $H^1_q$ is?

Question

I don't know the meaning of the notation $H^1_q$ (a Sobolev space)? Where I can find the exact definition of that notation? Thank you in advance.

Answer 1

A good resource for all things Sobolev-space related is Sobolev Spaces by R. A. Adams and J. J. F. Fournier, $2^\mathrm{nd}$ edition. Page $60$ there provides the definition that you're looking for:

$$H^m_q = H^{m,q}(\Omega) \equiv \mbox{the completion of } \{u\in C^m(\Omega) \mid \|u\|_{m,q} \lt \infty \} $$ where the completion is with respect to the norm $\|u\|_{m,q}$.

In fact, as was proven in a paper by Myers and Serrin with the famously short title H=W , the spaces $H^{m,q}$ and $W^{m,q}$ are the same so most of the time a choice between $H$ and $W$ will be entirely stylistic.