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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 $C^{1,1}$, this is true.

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.

Let $\mathring{H}^s$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^s$. 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.

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 $C^{1,1}$, this is true.

Let $\mathring{H}^s$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^s$. It is well-known that $$ [L^2(\Omega), \mathring{H^{2}}(\Omega)]_{\theta}=\mathring{H^{2s}}(\Omega) \quad \forall \theta\in[0,1]\, \text{and } \theta\neq \frac{3}{4} $$ 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.

Let $\mathring{H}^s$ be the closure of $\mathcal{D}(\Omega)$ under the norm $\mathring{H}^s$. 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.