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.


  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.
  • $\begingroup$ Why do you know that the given assertion is true? $\endgroup$ – Stefan Kohl Apr 11 '16 at 9:01
  • $\begingroup$ The statement was stated in a thesis, afterwards it was quoted in another paper. And that paper is cited many times. $\endgroup$ – Engineer Apr 11 '16 at 15:03
  • $\begingroup$ Then I suggest you to include the appropriate references in your question, to give it some context. $\endgroup$ – Stefan Kohl Apr 11 '16 at 15:08
  • $\begingroup$ Is $H^2$ here the $L^2$ Sobolev space with 2 derivatives? What are the norms $0,\Omega$ and $\frac12,\Omega$? $\endgroup$ – Willie Wong Apr 11 '16 at 15:10
  • $\begingroup$ @StefanKohl Thank you for your advice. I have added a link to the thesis. $\endgroup$ – Engineer Apr 11 '16 at 15:14

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