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Title page for ETD etd-05222012-212749


Type of Document Dissertation
Author LeCrone, Jeremy
URN etd-05222012-212749
Title On the axisymmetric surface diffusion flow
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Gieri Simonett Committee Chair
Akram Aldroubi Committee Member
Alexander Powell Committee Member
Emmanuele DiBenedetto Committee Member
Zhaohua Ding Committee Member
Keywords
  • fourier multipliers
  • nonlinear stability
  • maximal regularity
  • periodic boundary conditions
  • surface diffusion
Date of Defense 2012-05-04
Availability unrestricted
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.

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