At the end of the section of regularity regarding the elliptic equations (Chapter 6.3), Evans makes a comment in his Partial Differential Equations:
Would anybody explain what "energy" method means here? A quick search on Google returns that it might be related to the calculus of variations. But I don't see any calculus of variations in the mentioned section above.
If $u$ is the unknown of a PDE, expressions like $\int_U|u|^2$ and $\int_U|\nabla u|^2$ are usually interpreted in terms of energies related to the physical problem behind the PDE. Methods that involve estimates on that type of integrals are called energy methods. Historically, this may come from Hilbert's solution to the Dirichlet problem, which consisted in minimizing the integral $\int_U|\nabla u|^2$.