Ph.D. Thesis

Inserted: 26 feb 2024
Last Updated: 26 feb 2024

Pages: 83
Year: 2023
Links: https://iris.uniroma1.it/handle/11573/1682911

Abstract:

This thesis deals with regularity and rectifiability properties on the branching set of stationary varifolds that can be represented as the graph of a two-valued function. In the first chapter I briefly show the Simon and Wickramasekera’s work in which they introduce a frequency function monotonicity formula for two-valued $C^{1,\alpha}$ functions with stationary graph that leads to an estimate of the Hausdorff dimension of the branching set. In the second chapter I build upon Simon and Wickramasekera’s work and introduce several relaxed frequency functions in order to get an estimate of the Minkowski’s content of the branching set. I then use their result to prove the local $(n − 2)$-rectifiablility of the branching set.

Keywords: calculus of variations, Geometric measure theory, minimal surfaces