A feature of the mathematical physics of the 20th century is that there are a lot of researches devoted to the proof of the well-posedness of different boundary value problems and initial-boundary value problems. The proof of the well-posedness is usually based on a priori estimates for solutions in various norms. At present, mathematicians dealing with questions of wellposedness follow the rule to choose spaces that are natural for the problem under consideration.
One of the most important problems of mathematical physics was the problem (coming from the 19th century) of proving the well-posedness of the Dirichlet and Neumann problems for the Laplace equation and more general elliptic equations of second order, so that the proof must be based on the fact that the solutions to these problems are minimizer of the Dirichlet integral.