I have
$$\|u\|_{H^2(\Omega)} \leq \|\Delta u\|_{L_2(\Omega)} + \|u\|_{L_2(\Omega)} + \|u\|_{W^{3/2}_2(\partial\Omega_1)} + \|\nabla u \cdot n\|_{W^{1/2}_2(\partial\Omega_2)}$$
I somehow need to bound $\|u\|_{W^{3/2}_2(\partial\Omega_1)}$ and $\|\nabla u \cdot n\|_{W^{1/2}_2(\partial\Omega_2)}$ with the $L_2(\partial\Omega_1)$ and $L_2(\partial\Omega_2)$ norms.
As far as I understand this can not be done in an infinite dimensional sense.
Does there exists an inverse inequality that I can use to do that when $u$ is finite dimensional?