On the shape of the positivity region for a free boundary problem describing cell polarization

TL;DR

This paper analyzes a free boundary problem modeling cell polarization with small mass, showing convergence to an obstacle problem with an elliptical interface in the generic case.

Contribution

It proves convergence to a specific obstacle problem and explicitly characterizes the shape of the interface as an ellipse, including degenerate cases.

Findings

Solution converges to an obstacle problem in the plane.

The interface of the limit solution is an explicitly defined ellipse.

Degenerate maxima cases exhibit varied behaviors.

Abstract

In this paper we study a mass-constrained free boundary problem modeling cell polarization, in the regime where the mass is small. In the generic case of a signal with nondegenerate maxima, we prove that the solution converges locally to a global, integrable solution to an obstacle problem in the plane. We further show that the interface of the solution to the limit problem is an ellipse, the equation of which is explicit. We also study some cases where the signal has degenerate maxima, highlighting a variety of possible behaviors.