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On a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain and its computation

Eto, Tokuhiro Giga, Yoshikazu 北海道大学

2023.05.23

概要

In this study, we consider the mean curvature flow equation with prescribed contact angle
condition of the form
(
V = − divΓt n on Γt ∩ Ω for t ≥ 0,
(MCFB)
∠(n, nΩ ) = θ(t, ·) on ∂Γt ∩ ∂Ω for t ≥ 0,
where Ω ⊂ Rd is a smooth bounded domain and nΩ is the unit normal velocity vector field
on ∂Ω. Here, {Γt }t is a time evolving hypersurface to be determined and n represents the
outward unit normal vector field of Γt ; V denotes the velocity of Γt in the direction of n,
which is the outward unit normal vector to Γt , and θ is a given function on [0, T ] × ∂Ω that
describes the contact angle between Γt and ∂Ω for each t ≥ 0. Here, divΓt denotes the surface
divergence so that − divΓt n becomes the (d − 1 times) mean curvature of Γt in the direction
of n.
For the mean curvature flow equation in Rd , Almgren, Taylor and Wang [1] introduced a
time discrete approximation of the solution which is often called the Almgren-Taylor-Wang
scheme; a similar scheme is given by Luckhaus and Sturzenhecker [25]. ...

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