Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the Laplace norm $\Vert \Delta u\Vert^2_{L^2(\Omega)}$, first order norm $\Vert \nabla u\Vert^2_{L^2(\Omega)}$ and the boundary values $\Vert u|_{\partial \Omega}\Vert_{L^2(\partial\Omega)}$, $\Vert \partial_n u\Vert_{L^2(\Omega)}$?

So does there exist a $c\in \mathbb{R}_{>0}$ so that $$ \Vert D^2 u\Vert_{L^2(\Omega)}^2 \leq c\left( \Vert \Delta u\Vert^2_{L^2(\Omega)} + \Vert \nabla u\Vert^2_{L^2(\Omega)} + \Vert u|_{\partial \Omega}\Vert_{L^2(\partial\Omega)}^2 + \Vert \partial_n u\Vert_{L^2(\Omega)}^2\right) $$ for all $u \in H^2(\Omega)$? Is there a simple proof or literature I could look at? (Sadly, most of my books don't cover more than $H^1(\Omega)$ really when it comes to these estimations.)

I don't actually expect to need all of these values of $u$ but these are what I could afford to include if necessary. For a moment I was thinking about a "Poincaré-like" approach but this situation is maybe a little different.

Answer for the special case $(\Delta u)|_{\partial\Omega} = 0$

One possibility to partially solve this problem for the case $u \in H^2(\Omega)$ with $(\Delta u)|_{\partial\Omega} = 0$ would be to state that under certain assumptions there always exists an unique solution to the homogeneous Dirichlet problem $$ -\Delta w = f \\ w|_{\partial \Omega} = 0 $$ with $w \in H^1(\Omega)$ and $f \in L^2(\Omega)$. Given a domain $\Omega$ with sufficient regularity (in the literature this seems to be $C^2$-boundary, however there might be generalizations on that) we get $w \in H^2(\Omega)$ as well as the existence of a constant $c\in \mathbb{R}_{>0}$ (independent of $w$ and $f$) so that $$ \Vert w\Vert_{H^2(\Omega)}^2 \leq c \Vert f\Vert_{L^2(\Omega)}^2 $$ (see for example the book of Gilbarg and Trudinger mentioned by Behaviour in the comments). If you now choose $f:=-\Delta u \in L^2(\Omega)$ you get that the solution of the above equation is uniquely given by $w=u$ (since we have zero boundary values as of $(\Delta u)|_{\partial\Omega}=0$) and thus we have $$ \Vert u\Vert_{H^2(\Omega)}^2 = \Vert w\Vert_{H^2(\Omega)}^2 \leq c \Vert f\Vert_{L^2(\Omega)}^2 = c \Vert \Delta u\Vert_{L^2(\Omega)}^2. $$

So this estimate is mostly based on the regularity results you have for the Laplace equation for your domain and boundary values. Since I didn't find a regularity result for convex polygon-domains and inhomogeneous boundary conditions I restricted myself to that version until I find further results.

Also I want to mention that even if you had a regularity result for the inhomogeneous Dirichlet problem including a boundary term in form of $$ \Vert w\Vert_{H^2(\Omega)}^2 \leq c \left(\Vert f\Vert_{L^2(\Omega)}^2 + \Vert g\Vert_{L^2(\partial\Omega)}^2\right) $$ where $g \in L^2(\partial\Omega)$ describes the Dirichlet boundary condition on $\partial \Omega$, you'd end up with a result like $$ \Vert u\Vert_{H^2(\Omega)}^2 \leq c \left(\Vert \Delta u\Vert_{L^2(\Omega)}^2 + \Vert \Delta u \Vert_{L^2(\partial \Omega)}^2 \right), $$ which at least in my case includes information I don't have.

Now the question is if there exists a generalization of this result to the whole space $H^2(\Omega)$ without including any other terms than specified in the inequality at the beginning. Since it's important for applications one would also want to have the domain $\Omega$ as convex polygon at most.

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    $\begingroup$ Theorem 8.12 in the book by Gilbarg and Trudinger is of this kind. There is an awkward assumption there about having a function $\varphi$ in $W^{2,2}$ norm such that $u-\varphi\in W^{1,2}_0$... and the $W^{2,2}$ norm of $\varphi$ appears on the right hand side. I expect that for reasonable boundary values, you can take $\varphi$ to be the harmonic extension of the boundary values of $u$. $\endgroup$ – user147263 Dec 10 '14 at 2:39
  • $\begingroup$ @Behaviour Thanks, this book really is a valuable source! However, theorem 8.12 includes the $\Vert u\Vert_{L^2(\Omega)}$-norm, which I unfortunately can't access in my case, so it doesn't work with the exact problem I have. (At least it's not obvious to me how to maybe get rid of that term.) $\endgroup$ – Murp Dec 10 '14 at 18:37

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