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Let $\mathring{H}^\theta$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^\theta$. It is well-known that $$ [L^2(\Omega), \mathring{H^{2}}(\Omega)]_{\theta}=\mathring{H^{2\theta}}(\Omega) \quad \forall \theta\in[0,1]\, \text{and } \theta\neq \frac{3}{4} $$ when $\Omega$ is bounded and smooth. What is the interpolation result when $\Omega$ is Lipschitz continuous. What is the optimal regularity for $\partial\Omega$ to make such interpolation result hold.

It seems that if $\Omega$ is $C^{1,1}$, then this is true.

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  • $\begingroup$ @WillieWong Sorry. Actually, $\theta=s$ $\endgroup$
    – Ice sea
    Nov 27, 2017 at 14:18

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