Proving the Regularity of the Reduced Boundary of Perimeter Minimizing Sets with the De Giorgi Lemma

The Plateau problem consists of nding the set that minimizes its perimeter among all sets of a certain volume. Such set is known as a minimal set, or perimeter minimizing set. The problem was considered intractable until the 1960's, when the development of geometric measure theory by researchers such as Fleming, Federer, and De Giorgi provided the necessary tools to nd minimal sets. After the existence of minimal sets was proven, the study of perimeter minimizing sets became an active area of mathematical research{focused on determining the properties of minimal sets. One of the most prominent research problems sought to determine how smooth the boundary of minimal sets is in n dimensions, which is also known as the set's regularity. This paper approaches the study of perimeter minimizing sets using geometric measure theory, concluding on the De Giorgi Lemma{which demonstrates that minimal sets have some level of regularity.