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This paper is concerned with the motion of a time dependent hypersurface thatevolves by mean curvature flow with a a volume constraint. Phase fieldapproximation of this motion leads to the well known nonlocal Allen--Cahnequation. Here we propose a modified version of this equation, and we show thatit has better volume preserving properties than the classical one, even in thepresence of an additional forcing term.

A modified phase field approximation for mean curvature flow with conservation of the volume