A Theorem on Frequency Function for Multiple-Valued Dirichlet Minimizing Functions

TL;DR

This paper proves that for two-dimensional multiple-valued Dirichlet minimizing functions, the frequency function takes values of k/2, and characterizes their local behavior through blow-up analysis.

Contribution

It establishes a precise form for the frequency function in a special case and links it to the local behavior of the functions via blow-up analysis.

Findings

Frequency function is of the form k/2 for some nonnegative integer k.

Characterization of local behavior of functions through blow-up analysis.

Provides a specific structure for multiple-valued Dirichlet minimizing functions in 2D.

Abstract

This paper discusses the frequency function of multiple-valued Dirichlet minimizing functions in the special case when the domain and range are both two dimensional. It shows that the frequency function must be of value k/2 for some nonnegative integer k. Futhermore, by looking at the blowing-up functions, we characterize the local behavior of the original Dirichlet minimizing function.