M. Röger | Searcharxiv

arXiv2026

Localization properties of a free boundary problem for cell polarization

We investigate localization in a time-dependent contrained obstacle problem for the density of an active protein on a cell membrane. This free boundary problem has been derived in Logioti et al. [17] as a model for cell polarization under the influence of an external signal. We show that in the limit of small mass there is instantaneous localization, that is the mass of the active protein concentrates in the set where the signal is maximal for any positive time $t>0$. We give a precise a characterization of this concentration process on the appropriate slow time scale and illustrate this for a few examples of the external signal.

math.AP↗

arXiv2007

Convergence of phase-field approximations to the Gibbs-Thomson law

We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.

math.AP↗