Assume $L$ is an elliptic differential operator (second order, with coercive associated bilinear form) with smooth coefficients, and that $\Omega$ has smooth boundary.
Does there exist a result of the form
$$ \|u\|_{H^{k+2}(\Omega)}\leq C\|Lu\|_{H^k(\Omega)}+C\|u\|_{H^{k+1}(\partial\Omega)} $$
and where can I find it, in a better easily readable than most general form? I only find such results for homogenous boundary conditions.
See for example Theorem 2.3.3.6 in Grisvards "Elliptic problems in nonsmooth domains". Note that the boundary term as stated in the question is too optimistic, since the trace operator does not map $H^{k+2}(\Omega)\to H^{k+1}(\partial\Omega)$ but onto $H^{k+2-1/2}(\partial\Omega)$. The norm of this latter space is to be used (in particular, then all norms have the same scaling).