How one can identify the following (complex) interpolation space $$E_\theta :=[L^2(\Omega), H^2(\Omega)\cap H_0^1(\Omega)]_\theta,$$ where $\Omega$ is a regular domaine. After research, it seems that this depend on the position of $\theta \in (0,1)$:

for $0<\theta<1/4$, $E_\theta=H^{2\theta}(\Omega)$ and for $1/4 <\theta <1$ we have $E_\theta=H^{2\theta}_0(\Omega)$, while the case $\theta =1/4$ is critical.

Some inclusions are immediate while the others are not. Is there any elegant way to establish such identification?

Any reference would be helpful.