  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.
