| We develop a high-order nonlinear energy method to study the stability of steady states of the Stefan problem with surface tension. There are two prominent classes of steady states: flat planes and round spheres.;In the case of steady planes, we prove that the equilibria are always stable, asymptotically converging to a nearby flat hypersurface, in arbitrary dimensions. Our proof relies on an energy method along the fixed domain.;In the case of steady spheres, we establish sharp nonlinear stability and instability criterion in arbitrary dimensions. Our nonlinear stability proof relies on an energy method along the moving domain, and the discovery of a new "momentum conservation law". Our nonlinear instability proof relies on a variational framework which leads to the sharp growth rate estimate for the linearized problem, as well as a bootstrap framework to overcome the nonlinear perturbation with severe high-order derivatives. The instability result is surprising, as it stands in stark contrast to the known stability results for the related Mullins-Sekerka problem. |
