Is $[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}=H^1(\Omega)$?

Question

Is the interpolated space of order $\frac 1 2$,

$$[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}$$ equal to $H^1(\Omega)$?

I can't find any good examples of these interpolation ideas. Assume $\Omega$ is smooth.

Answer 1

Proposition 3.1 of Michael Taylor's PDE book (first volume) confirms that this is so for a compact manifold (which is what I was looking for).