Bounding the integral of an $H^1$ function on square domain

Question

I am currently trying to show that a linear functional is bounded so that I can use Lax-Milgram. The domain $\Omega$ is the square $0 \leq x,y \leq 1$, and I defined my Hilbert Space as $H = \{v \in H^1 : v = 0 \text{ on } \Gamma\}$, where $\Gamma$ is the left side of the square. Given how I defined my linear functional, I just need to show that there exists a $C_{\Omega}$ $$\| v \|_{L^2(\Gamma)} \leq C_{\Omega} \|v \|_{H^1(\Omega)}, $$ for all $v \in H$. The solution I'm reading just states that this constant exists, but I have no clue where it comes from, or if it is even valid. In general, can you bound the $L^2$ norm of a function over the domain's boundary by its $H^1$ norm on the interior?