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Let $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary $\Gamma$. Consider the following space $$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the norm $$\|u\|_{H_\Delta(\Omega)}^2 := \|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2,$$

Some authors define the previous space as $$H^1_\Delta(\Omega)=\{u\in H^1(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the norm $$\|u\|_{H^1_\Delta(\Omega)}^2 := \|u\|_{H^1(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2.$$

I'm wondering if the norms $\|u\|_{H_\Delta(\Omega)}$ and $\|u\|_{H^1_\Delta(\Omega)}$ are equivalent to $\|u\|_{H^2(\Omega)}$. What about the trace theorem in the spaces $H_\Delta(\Omega)$ and $H^1_\Delta(\Omega)$ ? Thank you.

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They not the same unless if you work with Dirichlet boundary conditions.

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  • $\begingroup$ Are they equivalent ?, I'm not working with Direchlet boundary conditions. $\endgroup$ – S. Maths Nov 10 '18 at 16:37
  • $\begingroup$ No, there is nothing which makes them equivalents. $\endgroup$ – Gustave Nov 10 '18 at 20:13
  • $\begingroup$ I need a regorous proof if possible. What about trace theorems over this space ? $\endgroup$ – S. Maths Nov 10 '18 at 21:35
  • $\begingroup$ I'm asking if the norms $\|u\|_{H_\Delta(\Omega)} := \|u\|_{L^2(\Omega)} +\|\Delta u\|_{L^2(\Omega)}$ and $\|u\|_{H^2(\Omega)}$ are equivalent. $\endgroup$ – S. Maths Nov 10 '18 at 21:53

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