# Interpolation for Sobolev spaces

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?

• @Hannes thank you, but this gives just the link to domains of fractional powers and not the explicit characterization depending on $\theta$. Aug 17 '20 at 19:28