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

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.