Glen Wheeler
University of Wollongong
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Featured researches published by Glen Wheeler.
Calculus of Variations and Partial Differential Equations | 2012
Glen Wheeler
We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH ≡ 0) hypersurface in
Bulletin of The Australian Mathematical Society | 2010
Glen Wheeler
Mathematische Annalen | 2013
James A McCoy; Glen Wheeler
{\mathbb{R}^3}
Mathematische Zeitschrift | 2011
James A McCoy; Glen Wheeler; Graham H. Williams
Transactions of the American Mathematical Society | 2014
Glen Wheeler
or
International Journal of Mathematics | 2013
Glen Wheeler
international workshop on digital watermarking | 2004
Glen Wheeler; Reihaneh Safavi-Naini; Nicholas Paul Sheppard
{\mathbb{R}^4}
Journal of Mathematical Analysis and Applications | 2011
Glen Wheeler
Discrete Mathematics | 2008
Martin W. Bunder; Keith Tognetti; Glen Wheeler
with restricted growth of the curvature at infinity and small total tracefree curvature must be an embedded union of umbilic hypersurfaces. Then we prove for surfaces that if the L2 norm of the tracefree curvature is globally initially small it is monotonic nonincreasing along the flow. We also derive pointwise estimates for all derivatives of the curvature assuming that its L2 norm is locally small. Using these results we show that if a singularity develops the curvature must concentrate in a definite manner, and prove that a blowup under suitable conditions converges to a nonumbilic embedded stationary surface. We obtain our main result as a consequence: the surface diffusion flow of a surface initially close to a sphere in L2 is a family of embeddings, exists for all time, and exponentially converges to a round sphere.
arXiv: Analysis of PDEs | 2014
Glen Wheeler
In this thesis the chief object of study are hypersurface flows of fourth order, with the speed of the flow varying from the Laplacian of the mean curvature, to the more general constrained flows which include a function of time in the speed, and satisfy various conditions. Our aim is to instigate a study of the regularity of these flows, answering questions of local and global existence, and some preliminary singularity analysis. Among our results are positive lower bounds for smooth and regular existence, classification of stationary solutions, interior estimates, and blowup asymptotics. Applying these results to a certain class of constrained surface diffusion flows, we obtain long time existence and exponential convergence to spheres for initial surfaces with small L norm of tracefree curvature. We present one application of this theorem, using it to deduce the isoperimetric inequality with optimal constant for 2-surfaces satisfying the above smallness condition. The theorem can be thought of as a stability of spheres result, as the smallness condition is an averaged distance from a standard round sphere to the initial manifold in L. This strengthens a related earlier result specialised to surface diffusion flow where the distance is small in C, obtained through a completely different method. The results throughout this thesis are new contributions for both surface diffusion flow, which has been considered by many authors, and the constrained flows, which have only recently been considered.