The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

Jeremy LeCrone; Yuanzhen Shao; Gieri Simonett

doi:10.3934/dcdss.2020242

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2020, Volume 13, Issue 12: 3503-3524. Doi: 10.3934/dcdss.2020242

This issue Previous Article A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs Next Article Large data solutions for semilinear higher order equations

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

1.
Department of Mathematics & Computer Science, University of Richmond, Richmond, VA 23173, USA
2.
Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA
3.
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

^* Corresponding author: Gieri Simonett
^* Corresponding author: Gieri Simonett

Received: December 31, 2018

Early access: January 2020

Published: August 2020

This work was supported by a grant from the Simons Foundation (#426729, Gieri Simonett)

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Abstract

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are $ C^{1+\alpha} $–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are $ C^{1+\alpha} $–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

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Mathematics Subject Classification: Primary: 35K55, 53C44; Secondary: 54C35, 35B65, 35B35.

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