I came across the expression
$$\Vert\nabla f\Vert_{L^2(\Omega)}$$
for some function $f: \Omega \subset \mathbb R^2 \to \mathbb R$, but I couldn't find the definition. Can anyone tell me how the $L^2$ norm of a gradient is defined?
My best guess is
$$\sqrt{ \int_\Omega |\nabla f|^2 dx}$$
where $|\nabla f|^2 = (\partial_{x_1} f)^2 + (\partial_{x_2} f)^2$, but I'm not sure whether this is correct.
Usually it is done the way you have suggested, because that way $L^2(\Omega,\mathbb{R}^2)$ (the space that $\nabla f$ lives in, when the norm is finite) becomes a Hilbert space.
There would be more choices though: If $\Vert\cdot\Vert_0$ is any norm on $\mathbb{R}^m$, then $L^p(\Omega,\mathbb{R}^m)$ inherits a norm by putting $\Vert (f_1,...,f_m)\Vert=(\int\Vert (f_1(x),\dots,f_m(x))\Vert_0^pdx)^{1/p}$, quite often one takes for $\Vert\cdot\Vert_0$ the corresponding $p$-norm on $\mathbb{R}^m$, as you have done in the case $p=2$ intuitively as well. However all norms obtained in that way are equivalent.
