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The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

  • * Corresponding author: Gieri Simonett

    * Corresponding author: Gieri Simonett

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

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  • 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.

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


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