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Let $\Omega \subset\mathbb{R}^d$ has smooth boundary and $$\Delta^k u \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega) \qquad k\geq0$$ Show that $u\in W^{n,2}(\Omega)$ for all $n\in \mathbb{N}$.

This is final step of problem 7.5.16 from Evans book.


What I tried:

For the start I want to show that $(\nabla^3 u)_{ijk} \in L^2$ for all $i,j,k$. Showing similar result for higher gradients should be similar.

I get into trouble even at the start, I don't know how to show that $\nabla^3 u$ even exist i.e. show that there is function $v\in L^1_{loc}(\Omega)$ that $$ \int_\Omega u \partial_i \partial_j \partial_k \phi = - \int v \phi \qquad \phi\in\mathcal{D}(\Omega) $$

After that I need to show that $\|\nabla^3 u\|_{L^2} < \infty$. For this I use inequality $\| \nabla^2 u \|_{L^2} \leq C \| \Delta u \|_{L^2}$. $$ \| \nabla^3 u \|_{L^2} =\| \nabla^2 \nabla u \|_{L^2} \leq C \| \Delta \nabla u \|_{L^2} = C\| \nabla \Delta u \|_{L^2} < \infty $$ Problem is that I do not know what are the exact conditions on $u$ such that used inequality $\| \nabla^2 u \|_{L^2} \leq C \| \Delta u \|_{L^2}$ holds.

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I know two methods to answer you question if the $\partial \Omega$ is regular enough. (1) You could obtain this result by Fourier transform. http://www.math.ucla.edu/~tao/254a.1.01w/notes2.dvi page10 (2) By the theory of regularity of the equation $-\Delta u = f$, $\|u\|_{H^2}\le \|f\|_{L^2}$.

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