Phase-field approximation of the Willmore flow

We investigate the phase-field approximation of the Willmore flow. This is a fourth-order diffusion equation with a parameter $\epsilon>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as $\epsilon$ trends to zero the level-set of solution will converge to motion by Willmore flow before the singularity of the later occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourth-order operator written as the sum of the squared Allen-Cahn operator and a singular linear perturbation.