Archive | 2019
Regularity of almost minimizing sets
Abstract
Resende, R. O. Regularity of almost minimizing sets. 2019. 120 f. Thesis (Master) Institute of Mathematics and Statistics, University of São Paulo, São Paulo, 2019. This work was motivated by the famous Plateau s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau s Problem, we make use of di erent theories: the theory of varifolds, currents and locally nite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau s problems. If we switch the problem to nd the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((λ, r)-minimizing sets) considered by Maggi ([Mag12]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition.