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

This paper is concerned with the motion of a time-dependent hypersurface ∂Ω(t) in ℝd that evolves with a normal velocity where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation where ε is an approximation parameter, W a double well potential and cW a constant that depends only on W. We study here a modified version of this equation and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume-preserving properties than the traditional nonlocal Allen–Cahn equation.