Proving the Regularity of the Reduced Boundary of Perimeter Minimizing Sets with the De Giorgi Lemma
PROVING THE REGULARITY OF THE REDUCED BOUNDARY OF PERIMETER MINIMIZING SETS WITH THE DE GIORGI LEMMA Presented by Antonio Farah In partial fulfillment of the requirements for graduation with the Dean's Scholars Honors Degree in Mathematics Honors, Department of Mathematics Dr. Stefania Patrizi, Supervising Professor Dr. David Rusin, Honors Advisor, Department of Mathematics THE UNIVERSITY OF TEXAS AT AUSTIN May 2020 Acknowledgements I deeply appreciate the continued guidance and support of Dr. Stefania Patrizi, who supervised the research and preparation of this Honors Thesis. Dr. Patrizi kindly dedicated numerous sessions to this project, motivating and expanding my learning. I also greatly appreciate the support of Dr. Irene Gamba and of Dr. Francesco Maggi on the project, who generously helped with their reading and support. Further, I am grateful to Dr. David Rusin, who also provided valuable help on this project in his role of Honors Advisor in the Department of Mathematics. Moreover, I would like to express my gratitude to the Department of Mathematics' Faculty and Administration, to the College of Natural Sciences' Administration, and to the Dean's Scholars Honors Program of The University of Texas at Austin for my educational experience. 1 Abstract The Plateau problem consists of finding 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 find minimal sets.
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