The surface diffusion flow for immersed hypersurfaces.

Joachim Escher, Uwe F. Mayer, Gieri Simonett

Abstract: The surface diffusion flow is given by the law that the normal velocity of the immersed hypersurfaces be equal to the Laplace-Beltrami of the mean curvature. This is a fourth order quasi-linear parabolic evolution problem. We show existence and uniqueness of classical solutions for the motion of an immersed hypersurface driven by this law (for any finite space dimension). If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical results displaying various phenomena of the surface diffusion motion, including the loss of embeddedness for initially embedded hypersurfaces, and the development of singularities.

1991 Subject Classification: 35R35, 35K55, 35S30, 65C20, 80A22

Key words and phrases: Surface diffusion, mean curvature, free boundary problem, immersed hypersurface, center manifold, maximal regularity, numerical simulation.

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