Scale Invariance of Sobolev Norms According to physical intuition, it seems that the Sobolev norm shouldn't be a physically useful quantity. Why is this the case, and why isn't a more dimensionally correct mixed norm employed more often?
Consider a function $f: \mathbf{R}^d \to \mathbf{R}$. If we denote the spatial units of $f$ by $A$, and the units of the output of $f$ by $B$, then an easy calculation shows that the units of $\| f \|_p$ is equal to $A B^{d/p}$. On the other hand, the output of the $i$th partial derivative $f_i$ has units $A/B$, which means that $\|f_i\|_q$ has units $AB^{d/q-1}$. From the perspective of a dimensional analysis, $\| f \|_p + \| f_i \|_q$ should only make sense if $AB^{d/p} = AB^{d/q-1}$.
Why, in this case, do we not consider the mixed norm Sobolev spaces  with $d/p = d/q-1$ more often than the standard Sobolev spaces $W^{1,p}(\mathbf{R}^d)$? Shouldn't the scale invariance make the mixed norm spaces more useful than the standard Sobolev spaces?
 A: So first, quantities are not always quantities with units! For example if you want to speek about a Gaussian $e^{-|x|^2} = \sum_{n=0}^∞ \frac{(-1)^n\,|x|^{2n}}{n!}$, it clearly has no homogeneity. So, it means that the $x$ has to be adimensional. Usually, in physical models, you will get something like $e^{-λ\,|x|^2}$ where if the unit of $x$ is $L$, then $\lambda$ is of unit $L^{-2}$. In partial differential equations, a lot of equations are about a density of something, which often have no units. At the end, it will depend a lot on your specific physical system behind, and the questions you are asking about it.
About Sobolev norms, you could as well just multiply by a factor and get $λ\,\|f\|_{L^p} + \|∇f\|_{L ^p}$. In part of the papers, one prefers to take $\lambda=1$ (of course if you have units it could be $λ = 1$ m/s for example, so the dimensional analysis is lost (but easy to modify usually).
Then, as you say, these Sobolev spaces are not homogeneous. One usually defines homogeneous Sobolev spaces $\dot{W}^{1,q}$ as the completion of $C^{\infty}_c$ with respect to the norm $\|∇f\|_{L^q}$, and defines the norm $\|f\|_{\dot{W}^{1,q}} = \|∇f\|_{L^q}$ on this space. This is homogeneous, so you can put directly units. Most of the other spaces of functional analysis have a homogeneous version.
Remark actually that your dimensional analysis is a good way to find the exponents for the Sobolev embeddings, since $W^{1,q} ⊂ L^p$ with your exponents. In particular, it implies that the norm $\|f\|_{L^p} + \|∇f\|_{L ^q}$ is equivalent to the homogeneous Sobolev space $\dot{W}^{1,q}$, since
$$
\|∇f\|_{L^q} ≤\|f\|_{L^p} + \|∇f\|_{L^q} ≤ (C+1)\,\|∇f\|_{L^q}
$$
so there is no need to add the $L^q$ norm, unless one really is interested by the value of the constant, or some other precise effects.
