Boundary Value Problems of Anisotropic Mean Curvature Flow

Mean curvature flow has been widely studied over the years. In this thesis, we focus on the boundary value problems of its anisotropic version. In particular, we study gradient estimates and the long-term behavior of solutions.

In Chapter 1, we give a brief overview of mean curvature flow and anisotropy. We also introduce the boundary value problems that we are interested in and summarize our main results.

Chapter 2 explores the properties of the anisotropic function, which is positive, convex, and homogeneous of degree one. We describe two conditions that ensure this function stays close to the isotropic case, allowing us to extend known results. We also examine the geometry and boundary data, which play a key role in later gradient estimates.

In Chapter 3, we study contact angle boundary problems for the anisotropic mean curvature flow in two dimensions. We provide a detailed proof of the gradient estimate, then show that solutions converge to a unique translating solution (up to translation). Although this approach does not extend to higher dimensions due to boundary complications, it works for the Dirichlet problem in any dimension, where similar results hold.

Chapter 4 deals with the higher-dimensional case. We design an auxiliary function related to the gradient and apply specific assumptions to handle the degeneracy of the anisotropic function. Using this setup, we establish gradient estimates and study the long-term behavior of solutions, showing their converge to translating solutions.

In Chapter 5, we apply a similar method to study the Neumann problem. We derive both gradient estimates and convergence results for its solutions.

Finally, Chapter 6 discusses possible future directions for research in mean curvature flow.