# An H2 estimate for Helmholtz equation

How to show the following statement?

Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,

$$-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \cdot n + i k u = g \quad\mbox{ on }\partial \Omega$$

Then the following estimate holds:

$$|u|_{H^2(\Omega)} \leq C \left(\Vert f+k^2 u \Vert_{0,\Omega} + \Vert\nabla u \cdot n\Vert_{\frac{1}{2},\partial \Omega}\right).$$

Thank you for any hint on this.

Remark:

1. This statement is quoted from a thesis On generalized finite element methods by Jens Markus Melenk on P.121. It says using regularity theory for '$-\Delta$' to get the statement.
2. I made a mistake. The requirement for $\Omega$ should be bounded convex, not bounded Lipschitz.
• Why do you know that the given assertion is true? – Stefan Kohl Apr 11 '16 at 9:01
• The statement was stated in a thesis, afterwards it was quoted in another paper. And that paper is cited many times. – Engineer Apr 11 '16 at 15:03
• Then I suggest you to include the appropriate references in your question, to give it some context. – Stefan Kohl Apr 11 '16 at 15:08
• Is $H^2$ here the $L^2$ Sobolev space with 2 derivatives? What are the norms $0,\Omega$ and $\frac12,\Omega$? – Willie Wong Apr 11 '16 at 15:10