On the axisymmetric surface diffusion flow

Abstract

In this thesis, we establish analytic results for the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In the first part of the work, we develop a general theory establishing maximal regularity results for a broad class of abstract, higher-order elliptic operators, in the setting of periodic little-Hölder spaces. These results are then applied, in the second part of the thesis, to prove well-posedness results for ASD. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + alpha)-little-Hölder regular surfaces of revolution embedded in R^3. Further, we give conditions for global existence of solutions and we prove that solutions are real analytic in time and space. We also investigate the dynamic properties of solutions to ASD in the second part of the thesis. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. We focus on the family of cylinders as equilibria of ASD and we establish results regarding the stability, instability and bifurcation behavior of cylinders with the radius acting as a bifurcation parameter for the problem.