What is the norm in the interface space $L^2(\Gamma)$? Given bounded open domain $\Omega \subset \mathbb{R}^2=\{x={(x_1,x_2):x_i\in \mathbb{R}}\}$. $\Omega$ is divided by an interface $\Gamma$ into 2 open subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega=\Omega_1\cup\Gamma\cup\Omega_2$ and $\Omega_1\cap\Omega_2=\emptyset$. What are the definition of space $L^2(\Gamma)$ and its norm?

Generally, $L^2(\Gamma)$ means the Lebesgue space with respect to some measure $\mu$ on the set $\Gamma$. When the measure is not specified, it is to be inferred from the context or from the structure of $\Gamma$. On some sets, e.g., a sphere, there is a "preferred" measure that is used without a specific mention. 
Here I would assume that $\mu$ is the restriction of the $1$-dimensional Hausdorff measure to $\Gamma$, which in less technical terms is known as "arclength measure". Integration with respect to this measure is carried out by parametrizing $\Gamma$ by an absolutely continuous function $\gamma:[0,1]\to\Gamma$ and changing the variable:
$$
\int f_\Gamma = \int_0^1 f(\gamma(t))\|\gamma'(t)\|\,dt
$$
