Stability of Serrin's Problem and Dynamic Stability of a Model for Contact Angle Motion

TL;DR

This paper establishes a new linear stability result for Serrin's symmetry problem, linking it to a dynamic model of contact angle motion, and demonstrates exponential convergence to equilibrium, bridging geometric stability and dynamic behavior.

Contribution

It introduces a novel stability estimate for Serrin's problem based on a weak norm, connecting static geometric stability with the dynamic stability of contact angle motion.

Findings

Proves a linear stability estimate depending on the L^2(\u2206  ext{ boundary norm} ) norm.

Shows exponential convergence to steady state for the dynamic contact angle model.

First application of geometric minimization stability to dynamic stability of a gradient flow.

Abstract

We study the quantitative stability of Serrin's symmetry problem and it's connection with a dynamic model for contact angle motion of quasi-static capillary drops. We prove a new stability result which is both linear and depends only on a weak norm \[ \big\||Du|^2- 1\big\|_{L^2(\partial \Omega)}. \] This improvement is particularly important to us since the $L^2(\partial \Omega)$ norm squared of $|Du|^2-1$ is exactly the energy dissipation rate of the associated dynamic model. Combining the energy estimate for the dynamic model with the new stability result for the equilibrium problem yields an exponential rate of convergence to the steady state for regular solutions of the contact angle motion problem. As far as we are aware this is one of the first applications of a stability estimate for a geometric minimization problem to show dynamic stability of an associated gradient flow.