Following Grisvard, if $\Omega$ is an open, bounded and convex domain, and $f\in L^2$, there exist a unique $u\in H^2$ solution of $$ \left \{ \begin{aligned} -\Delta u&=f\; \textrm{in } \Omega, \\ u&=0\; \textrm{in } \partial \Omega. \end{aligned} \right. $$
Now, I would like to know if there is a similar result for the following problem $$ \left \{ \begin{aligned} -\Delta u&=f\; \textrm{in } \Omega, \\ u&=0\; \textrm{on } \Gamma, \\ \frac{\partial u}{\partial n}&=0\; \textrm{on } \partial \Omega \setminus\Gamma, \end{aligned} \right. $$
where $\Gamma$ is an non-empty subset of $\partial \Omega$. I know that with Lax-Milgram theorem, one can easily get the $H^1$ regularity. But, is it possible to have an $H^2$ regularity as in the previous case ?
Thanks in advance for the answer.