Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$. I have found in several research articles that if a function $u \in L^2(\Omega)$, $\nabla u \in L^2(\Omega)$ and $\Delta u \in L^2(\Omega)$ then $u \in H^2(\Omega)$. How does this conclusion hold? what absorbs the mixed partial derivative?
Please help me with references.
As Jochen mentioned, this is a consequence of regularity for the Poisson equation, at least for a regular domain ($C^2$ would suffices, I don't know if one can take less regularity such as $C^{1,1}$ or $C^{1,\alpha}$.).
Indeed, if one set $f = -\Delta u$ and $g = u_{|\partial \Omega}$ then by definition of the trace and the fact that $u \in H^1(\Omega)$, one has $f \in L^2(\Omega)$ and $g \in H^{1/2}(\partial \Omega)$ (i.e. the image of $H^1$ under the trace operator). Then regularity gives you $u \in H^2(\Omega)$ (see thm 8.12 of Elliptic partial differential equations of second order by D. Gilbard and N. S. Trudinger).
Note that the assumption $\nabla u \in L^2(\Omega)$ can't be dropped for general domain, see this question, except when $\Omega = \mathbb{R}^d$ (or in the torus) where one can use Fourier transform to obtain easily the result.
