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Let $ \Omega $ be an open subset with compact closure of a Riemannian manifold $ M $. Let $ u \in H^1_{0}(\Omega) $ be a weak solution of the Dirichlet boundary problem:

$$ -\Delta u + qu = f \; \; \textrm{in} \; \Omega \; \; \; (1) $$

$$ u|_{\partial\Omega}=0 $$

In other words $ u $ satisfies the following equality for every $ \psi \in C^{\infty}_{0}(\Omega) $

$$ \int_{\Omega} \langle \nabla u,\nabla \psi \rangle + q u \psi = \int_{\Omega} f\psi \; \; \; (2)$$

Now from standard regularity results if $ q,f \in C^{\infty}(M) $ then $ u \in C^{\infty}(\Omega) $ but in general $ u \notin C^{\infty}(\overline{\Omega}) $. If $ \partial \Omega $ is smooth then $ u \in C^{\infty}(\overline{\Omega}) $ (and since $ u \in H^1_{0}(\Omega) $ we have in particular that $ u \in C^{\infty}_{0}(\overline{\Omega}) $ ) and by an integration by parts on (2) we can recover that u in actually a classical solution of (1) (in other words $ u $ satisfies equation (1) pointwise).

But for a general $\partial \Omega $ we have that $ u $ can have singularities on the boundary and we cannot apply the integration by parts. So i'm asking if it is true that in this case $ u $ is a classical solution of (1). Is it? The question seems me interesting since in this case we know that $ u \in C^{\infty}(\Omega) $.

Thanks

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  • $\begingroup$ I think that the test space must be $C_0^\infty(\Omega)$. $\endgroup$ – Tomás Jul 4 '13 at 17:51
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    $\begingroup$ What is your definition of classical solution? $\endgroup$ – Tomás Jul 4 '13 at 18:34
  • $\begingroup$ Yes you're right. I'hve corrected the mistake. $\endgroup$ – user55449 Jul 5 '13 at 8:08
  • $\begingroup$ I've edited the answer. I hope that now it is more clear. $\endgroup$ – user55449 Jul 5 '13 at 8:16
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Assume that $u\in C^{\infty}(\Omega)$. Consider any open set $U\subset\Omega$ such that $\overline{U}\subset\Omega$ and $U$ has nice boundary. We have that $$\int_U\nabla u\nabla\psi+\int qu\psi=\int f\psi,\ \forall\ \psi\in C_0^\infty(U)$$

In particular, by appying Green formula, we get that $$\int\left(-\Delta u+qu-f\right)\cdot\psi=0,\ \forall\ \psi\in C_0^\infty(U)$$

The last equality implies that $-\Delta u(x)+q(x)u(x)=f(x)$ for all $x\in U$. For each fixed point $x\in \Omega$, we take $U=B_r(x)$ where $r>0$ is choosen in such a way that $\overline{B_r(x)}\subset\Omega$. we concude from the previous argument that $$-\Delta u(x)+q(x)u(x)=f(x),\ \forall\ x\in\Omega$$

To conclude, we have to prove that $u(x)=0$ in $\partial \Omega$. But to this happens, we need some regularity on the boundary of $\Omega$, for example, $\partial\Omega\in C^{0,1}$ is sufficient to conclude that $u\in C(\overline {\Omega})$. Indeed, the fact that $f\in C(\overline{\Omega})$ implies that $f\in L^p(\Omega)$ for all $p\in (1,\infty)$, therefore, $u\in W^{1,p}(\Omega)$ for all $p>1$. In particular, by using Sobolev embedding, we can conclude that $u\in C(\overline{\Omega})$, which would implies by the trace theorem that $u(x)=0$ in the boundary.

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