Let $L$ be a uniformly strongly elliptic second order differential operator with coefficients in $W^{k+1,\infty}$, $f \in H^{k-1}(\Omega)$ and $u_{|\partial \Omega} \in H^{k+1/2}(\partial \Omega)$. Consider the problem:
$$ \begin{split} Lu &= f \quad \text{in } \Omega\\ u &= g \quad \text{on } \partial \Omega, \end{split} $$
can someone tell me if this estimate is true?
$$\|u\|_{H^{k+1}(\Omega)} \leq C\left(\|f\|_{H^{k-1}(\Omega)} + \|g\|_{H^{k+1/2}(\partial \Omega)} \right)$$
especially for $k=1$:
$$\|u\|_{H^2(\Omega)} \leq C\left(\|f\|_{L^2(\Omega)} + \|g\|_{H^{3/2}(\partial \Omega)} \right)$$