Nonlinear stability results for the modified Mullins–Sekerka and the surface diffusion flow

It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins–Sekerka or Hele–Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta–Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins–Sekerka flow.