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My question is simple: Let $\Omega$ be a bounded smooth domain. Does the following inequality $$\|D^2w\|_{L^q(\Omega)}\leq C\|\Delta w\|_{L^q(\Omega)},$$ hold for any function $w\in W^{2,p}(\Omega)\cap C^1(\overline \Omega)$, such that $\frac{\partial w}{\partial \eta}=0$ on $\partial \Omega$, where $\eta$ denotes the outward normal vector, $C$ is a positive constant, $D^2w$ denotes the Hessian of $w$, and $\Delta w$ is the Laplacian of $w$?

I am familiar with the theory in the local case and under Dirichlet boundary condition.

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Yes, it's true. Check Theorem 2.3.3.6 of the book

P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985.

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  • $\begingroup$ Thank you for the reference. P.S. Felice Iandoli sends his regards $\endgroup$
    – Domenico Vuono
    Commented Jul 22 at 10:13
  • $\begingroup$ Piacere. E salutami lo ! $\endgroup$
    – Ayman Moussa
    Commented Jul 23 at 15:39

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