What are the examples of *practical* applications of Sobolev spaces?

The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of existence, stability and regularity of solutions, convergence of numerical methods etc. But what are the examples of *less theoretical* uses of Sobolev spaces?
In particular, what are the examples of

- PDEs whose explicit solution can only be found using the theory of Sobolev spaces
- problems when Sobolev spaces are an essential ingredient of otherwise impossible computation of some physically relevant quantity (heat flux, drag force etc.)
- PDEs from Physics (or other areas) when the solutions are not smooth but are Sobolev (by nature of the problem).

Remark. Maybe this has to be a community wiki question.