TITLE: Why are Sobolev spaces useful?
QUESTION [26 upvotes]: Why are Sobolev spaces useful, and what problems were they developed to overcome? I'm particularly interested in their relation to PDEs, as they are often described as the 'natural space in which to look for PDE solutions' - why is this? How do weak solutions and distributions come into this?
There are plenty of books on the subject, but these seem to jump straight into the details and I'm struggling to see the big picture. I know that the Sobolev norm makes the function spaces complete, which would guarantee that infinite linear combinations of solutions do not leave the space, as can be a problem when working with $\mathscr{C}^2$, for example, but are there any other reasons why this norm is important?
I'm also interested in the Sobolev embedding theorems, since I believe that they're important in the problems I'm trying to solve. These are (1) proving the compactness of the integral operator whose kernel is the Green's function for the Laplacian on a bounded domain $\Omega \subset \mathbb{R}^{n}$ with smooth boundary, and (2) understanding why minimising functions of the Rayleigh quotient,
$${\arg\min}_{f \in T} \int_{\Omega} \frac{\nabla f \cdot \nabla f}{\left< f , f \right>}$$
always exist, and that they are necessarily smooth ($\mathscr{C}^\infty(\Omega)$) among the set of trial functions $T$ of continuous functions with piecewise continuous first derivatives which vanish at the boundary and are not identically zero. To me, this sounds like the Sobolev space $H_0^1 (\Omega)$ at work, where the smoothness is the result of a Sobolev embedding theorem; however, I'm very new to Sobolev spaces and so don't know much about this. Could anyone provide me with some insight into how results (1) and (2) might be proven?

REPLY [3 votes]: In addition to the previous answer, I think it is useful to note that spaces $C^k[a,b]$ and such are Banach spaces, with norms given by the sum of the sup-norms of derivatives up to order $k$. So "having Banach spaces" does not entirely explain why we might want Sobolev spaces.
In my experience, using $L^p$ norms (especially $L^2$) of derivatives up to order $k$ is just technically better than using sup-norms of pointwise values. For one thing, it seems to be that pointwise values are very fragile, and not really to-the-point, so, actually, dodging "almost-everywhere" issues is quite sane.
Also, already in 1906-7, G. Levi and then G. Frobenius used the $L^2$ "Sobolev space" $H^1$, to prove a correct version of "the Dirichlet principle", which asserts a minimum principal for spaces of continuous, etc., functions... which as originally naively conceived is simply false. The implicit point is that in Hilbert spaces it is true that a closed, convex set has a unique element of minimal norm (as opposed to none or infinitely-many), so essentially all the applications of the (originally naive/false) form of Dirichlet's principle were recovered via this variant.