Relaxation rates for the Mullins-Sekerka evolution of nearly circular curves in the plane

In this talk, we consider the two-phase Mullins-Sekerka evolution of nearly circular simple closed curves in the plane. Transferring the method of O. Chugreeva, F. Otto and M. G. Westdickenberg from 2019 to graphs over equilibrium configurations with non-trivial background curvature, we capture and quantify the respective relaxation rates. Our result establishes the existence of two qualitatively different relaxation regimes, an initial regime of (lengthscale-independent) algebraic decay and a latter regime in which (lengthscale-dependent) exponential decay takes over. This talk is based on joint work with M. G. Westdickenberg and U. Hryniewicz.