Let $L=\nabla\cdot(A \nabla )$ be a second order partial differential operator whose smooth coefficient matrix $A$ is positive-definite. Then for compactly supported smooth $u$ it holds for $s=2$
$\Vert u\Vert_s \leq C(\Vert Lu\Vert +\Vert u\Vert)$
where $\Vert \cdot \Vert$ is simply the $L^2$ norm and $\Vert \cdot \Vert_s$ is the fractional Sobolev space norm of order $s$.
I have two questions:
1) Does $\Vert \cdot \Vert_s$ equal the Sobolev space norm $\Vert \cdot\Vert_{W^{1,2}}$ when $s=2$ or are they at least equivalent in the sense of norms? I am asking this, because I have seen the above result stated for $W^{1,2}$ but not not for fractional Sobolev space norms.
2)As mentioned above, I have seen this result stated in the $W^{1,2}$ norm but then only for uniformly elliptic operators. Does it also hold for only elliptic operators and if yes, what is a good reference?