# H_0^1 and C_infinity on the interior, does that imply classical limit is 0 on the boundary?

The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H_0^1 and also belong to C_infinity on the interior. Does that mean the classical limit of the function on the boundary is 0?

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What if we only know that the function is C_infinity on the open interior? Would the classical limit still be 0? – John Zheng Mar 24 '13 at 17:44

Yes. By Theorem 9.17 in Brezis' book, let $\Omega$ be just a bit regular$^*$ and let $u\in W^{1,p}(\Omega)\cap C(\bar{\Omega})$. Then $u\in W^{1,p}_0(\Omega)$ if and only if $u(z)=0$ for all $z\in \partial \Omega$.
$^*$ $C^1$ will do, but even less regularity might be really necessary, I guess.
Sorry, I just realized that there is a bug (?) in MO-code so that it will accept \bar but not \overline. This induced a wrong version of Brezis' result. You can now see the actual version. Concerning your new question: Your setting should be formulated in a more precise way (what kind of elliptic problem are you thinking of?) Anyway, relatively mild assumptions imply that the solution is in $C^2(\bar{\Omega}$, cf.Thm. 9.25 in Brezis. (Btw: for better later readability you should in the future highlight your edits, otherwise the first answers might look bizarre/off-topic/wrong afterwards) – Delio Mugnolo Mar 24 '13 at 21:39