Abstract: Direct methods in the Calculus of Variations provide existence of minimizers under compactness and lower semicontinuity in suitable function spaces and topologies. Next, regularity techniques serve to show that weak solution are indeed classical ones. Therefore, regularity theory is a crucial step in order to get solvability.
I will give an overview passing through the fundamental contributions of De Giorgi, Nash, Moser, Campanato, Uhlenbeck right up to problems with general growth or double phase.