Under what conditions does $u$ with $supp(u) \subset \Omega$ belong to $H_0^1(\Omega)$?

Question

The question is pretty simple. Suppose that $\Omega \subset \mathbb{R}^n$ is bounded and that $u \in H^1(\Omega)$.

Under what conditions does $u$ with $supp(u) \subset \Omega$ belong to $H_0^1(\Omega)$?

We already know that if $u $ is continuous then, as $u=0$ on $\partial \Omega$, $u \in H_0^1(\Omega)$. We also know that if $p>n$ then $u$ is continuous (i.e. $u$ has a continuous representative) and the result follows. In this case $p=2$ so this trick only works for an interval $I \subset \mathbb{R}$.

It is somewhat intuitive that $u$ should belong to $H_0^1(\Omega)$, at least with a weaker hypotesis than $u \in C(\Omega)$.

Answer 1

Suppose $\partial\Omega$ is $C^1$. Note that $W^{1,2}(\Omega)=H^1(\Omega)$.

The following two theorems (see Partial Differential Equations (chapter 5) by Evans) can answer your question:

Answer 2

As clarified in the comments, we have $$\{ x \in \Omega \mid u(x) \ne 0 \} \subset V \subset \Omega,$$ where $V$ is a compact set. Hence, $V$ has a positive distance $\delta > 0$ to the boundary $\partial\Omega$. Mollifying $u$ with a mollifier with radius smaller than $\delta$ produces a sequence of smooth functions in $C_0^\infty(\Omega)$ converging towards $u$ in $H^1(\Omega)$. Hence, $u \in H_0^1(\Omega)$.