Mass conserving Allen-Cahn equation and volume preserving mean curvature flow

摘要

We consider a mass conserving Allen-Cahn equation u_t = △u + ε~(-2)(f(m)-ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(·, t)) and-f is the derivative of a double equal well potential. Given a smooth hypersurface γ_0 contained in the domain, we show that the solution u~ε with appropriate initial data tends, as ε ↘ 0, to a limit which takes only two values, with the jump occurring at the hypersurface obtained from the volume preserving mean curvature flow starting from γ_0.