In their paper The spectral drop problem, Buttazzo and Velichkov state that the embedding $H_0^1(\Omega, D) \hookrightarrow L^2(D)$ is compact, where $\Omega \subset D$ is quasi-open, $D \subset \mathbb R^N$ is a Lipschitz open domain and $|\Omega| < |D|$ and $$ H_0^1(\Omega, D) = \{u \in H^1(D) \ : \ u = 0 \text{ quasi-everywhere in } D \setminus \Omega\}. $$

They claim that this is a well-known fact and follows from the Lipschitz condition on $D$, but I wasn't able to find a proof.

At first I thought about adapting the classical proof of the Rellich-Kondrachov theorem, but I failed.

Can you recommend any reference where I can find a detailed proof of this fact, or any hints?

I am beginning to learn about capacities and quasi-things, and would like to see the proof to get more acquainted with the theory.

Thanks in advance.

EDIT

Related notions

The capacity of a set $A \subset \mathbb R^N$ is $$ \text{cap}(A) = \inf\left\{\int_{\mathbb R^N} (|\nabla u|^2 + u^2) \ dx \ : \ u \in H^1(\mathbb R^N), \ u \geq 1 \text{ in a neighborhood of } A\right\} $$

A set $\Omega \subset \mathbb R^N$ is quasi-open if, for every $\varepsilon > 0$, there exists an open set $\omega_\varepsilon$ such that $\text{cap}(\omega_\varepsilon) \leq \varepsilon$ and $\Omega \cup \omega_\varepsilon$ is open.

A property is said to hold quasi-everywhere if it holds everywhere except, at most, in a set of zero capacity.