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While working on regularity theory for variational formulation of PDEs I often find ''Let $\Omega \subset \mathbb R^n$ and $u$ be sufficiently regular so that Green's identity $$ \int_\Omega (-\Delta u)v = \int_\Omega \nabla u \nabla v - \int_{\partial \Omega}v \partial_n u$$ holds''. My question is: How regular does $\Omega$ has to be so that the above statement holds for $u \in H^2 (\Omega)$?

Thanks so much in advanced,

D

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$\partial\Omega$ and $u$ have to be smooth enough in order for the integral $\int_{\partial\Omega}v\partial_n u$ to make sense.

What suffices is: Lipschitz continuity for the boundary and $\partial_n u\in L^2(\partial\Omega)$, for $u$. Even with weaker assumptions it is OK, for example $\partial_n u\in H^{-1/2}(\partial\Omega)$.

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  • $\begingroup$ The reference I had in mind (PDEs by Avner Friedman) is not reader friendly book. Let me check and see if I can find anything friendlier. $\endgroup$ – Yiorgos S. Smyrlis Jan 5 '14 at 16:43
  • $\begingroup$ I looked but could not find references to regularity. I found the theory of adjoint equations, which amounts to the same thing, but it was all about smooth functions. $\endgroup$ – D G Jan 5 '14 at 17:01
  • $\begingroup$ See Theoretical Numerical Analysis: A Functional Analysis Framework By Kendall Atkinson, Weimin Han - Chapter 8. $\endgroup$ – Yiorgos S. Smyrlis Jan 5 '14 at 18:24
  • $\begingroup$ Found it. Thank you so much for your effort. $\endgroup$ – D G Jan 5 '14 at 19:33

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