I am wondering what's the correct way to mathematically describe the following problem. Say you have an object that can be defined as an open set $\Omega \in \mathbb{R}^d$, where the dimension $ d=2,3$. What you normally find in scientific articles on mechanics is that the boundary of the object is obtained as $\Gamma \equiv \overline{\Omega} \setminus \Omega$ and then you can define a vector-valued function space in $\Omega$ as $$ \mathcal{V} = \left\{ \left. \mathbf{v} : \mathbf{v} \left( \mathbf{x} \right) \in \mathbb{R}^d \right| \, v_i \in H^1 \left( \Omega \right), i=1,\ldots,d \right\}, $$ where $H^1 \left( \Omega \right)$ is the Hilbertian Sobolev space of square integrable functions (and square integrable first-order derivatives).
So far so good, but now let's say you have $n$ sets of measure zero within $\Omega$, say $\Gamma_i^c, i=1,\ldots, n$. In my case $\Gamma_i^c$ represents a crack in the object of the body where clearly the function is not defined (the function $\mathbf{v}$ has a jump). But I can still compute the integral of $\mathbf{v}$ at both sides of the crack by properly dividing the integration regions (so the crack has measure zero). This now confuses me as I don't know how to properly define my vector-valued function space.
Can I just simply state $$ \mathcal{V} = \left\{ \left. \mathbf{v} : \mathbf{v} \left( \mathbf{x} \right) \in \mathbb{R}^d \right| \, v_i \in H^1 \left( \Omega \setminus \cup_j \Gamma_j^c \right), i=1,\ldots,d \right\} $$ or this is wrong? Or is it that in this case $\Gamma$ takes not only the outer boundary but also the internal cracks? What's the proper way to mathematically describe the vector-valued function space for this problem?