Unlike their Holder cousins, most Sobolev spaces are *reflexive* Banach spaces. Reflexivity is a highly desirable feature for variational problems because it gives a little bit of compactness enough so you can prove the existence of minimizers (or more general critical points) of various energy functionals. Such critical points satisfy the Euler-Lagrange equations and thus you obtain existence of (weak) solutions of many important equations in geometry or physics. Given that their norms have an integral description, the Sobolev spaces are tailor made for energy functionals described by integrals. The famous and for a while controversial Dirichlet principle states that any function $u$ defined on a compact smooth domain $\newcommand{\bR}{\mathbb{R}}$ $\Omega\subset \bR^N$ which is zero on the boundary of $\Omega$ and minimizes the energy functional

$$E(u)=\int_\Omega\left(\frac{1}{2}|\nabla u(x)|^2 -f(x) u(x)\right) dx $$

must be a solution of the Poisson problem $\Delta u=f$ in $\Omega$ satisfying the boundary conditions $u=0$ on $\partial \Omega$.

Weierstrass pointed out a major flaw in the classical understanding of this principle by constructing, in a special case, a minimizer of this energy functional which is not twice differentiable so the Laplacian does not make sense.

That stopped things in their tracks for a while until Hilbert, in his famous 1900 Paris Conference talk included this in the list of his famous 27 problems. In particular, he hinted to a way out by stating that any variational problem has a solution provided that, if need be, we suitably define the concept of solution.

You can read more about this and see many applications of Sobolev spaces to geometry in these lectures.

Some Nonlinear Problems in Riemannian Geometry. – Ben McKay Mar 5 at 16:43Sobolev Spaces on Riemannian Manifolds. – Ben McKay Mar 5 at 16:46