The 1-dimensional [lambda]-self shrinkers in R² and the nodal sets of biharmonic Steklov problems

Abstract

This thesis contains two of my projects. Chapter 1 and 2 describe the behavior of 1-dimensional [lambda]-self shrinkers, which are also known as [lambda]-curves in other literature. Chapter 3 and 4 focus on the estimation of the asymptotic behavior of the nodal set of biharmonic Steklov problems. Chapter 1 gives the background of mean curvature flow and the importance of self-shrinkers as solitons of the flow equation. We also introduce the background of the [lambda]-hypersurface and explain how this is related to the self shrinkers. In chapter 2, we examine the solutions of 1-dimensional [lambda]-self shrinkers and show that for certain [lambda] < 0, there are some closed, embedded solutions other than circles. For negative [lambda] near zero, there are embedded solutions with 2-symmetry. For negative [lambda] with large absolute value, there are embedded solutions with m-symmetry, where m is greater than 2. Chapter 3 focuses on the background of spectral geometry. Several eigenvalue problems are introduced. We have a brief survey of some of the important problems such as the asymptotic distribution of the eigenvalues, the shape optimization problem and the bound of nodal sets. This project focuses on establishing a lower bound of the measure of the nodal set. In chapter 4, we use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.