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I often read this:

Let $\Omega$ be a open bounded set. There is a unique $u \in H^1(\Omega)$ such that $$-\Delta u = f \text{ on $\Omega$}$$ $$u|_{\partial \Omega} = g$$

But how can we write $u|_{\delta \Omega}$ when $\Omega$ is an open set so doesn't contain its boundary??? Note that I am not asking about the trace operator. I am not asking how we can specify something on a null set. I am asking why we can say that $$u(\text{some boundary point})$$ makes sense when the boundary is not in $\Omega$ and we only know that $u \in H^1(\Omega)$, not $u \in H^1(\overline \Omega).$

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I think people use this rather vague statement just to avoid presenting extra technicalities in Sobolev spaces theory. The full picture should be:

  • Restriction on boundary is valid for any $C^{\infty}(\overline \Omega)$. Hence we define the trace: $T:C^{\infty}(\overline \Omega)\rightarrow L^2(\partial \Omega) $.

  • $C^{\infty}(\overline \Omega)$ is dense in $H^1(\Omega) = W^{1,2}(\Omega)$, hence we have a continuous linear extension $\widetilde{T}:H^1(\Omega) \rightarrow L^2(\partial \Omega)$.

  • The trace of any $u \in H^1(\Omega)$ should be: consider $\{u_n\}\subset C^{\infty}(\overline \Omega)$, let $u_n\to u$ under $H^1$-norm, define $$ u|_{\partial \Omega} = \lim_{n\to \infty}Tu_n $$

Lastly, we could not define $$u(\text{some boundary point})$$ the trace operator is rather a densely defined operator (well-defined on a dense subset).

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It's possible that $u \in H^1(\Omega)$ isn't even continuous on $\Omega$, so it's not even clear what we'd mean by $u(\text{some interior point})$. But if $\partial\Omega$ is $C^1$, then the trace of $u$ on $\partial\Omega$ is a well-defined function in $L^2(\partial\Omega)$, and this is what we mean when we write $u|_{\partial\Omega}$. No claim is being made about any pointwise values.

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