The singular set in the Stefan problem
The classical Stefan problem, first introduced in 1831 by Lamé and Clapeyron, aims to describe the evolution of the temperature in a block of ice (initially at zero degrees centigrade) that is melting to water. The Baiochi-Duvait transformation reduces it to the parabolic obstacle problem, easier to treat in many aspects. It is known since the celebrated work of Caffarelli that the interface between ice and water (the so-called free boundary) is smooth outside of a closed set of the spacetime, which is called the singular set.
In the talk, I will introduce a forthcoming joint work with A. Figalli and X. Ros-Oton on the size and regularity of this singular set.
In three spatial dimensions we prove that the singular set splits into two pieces. The first is a set of parabolic dimension 1 (hence it is small). The second piece satisfies that its spatial projection can be covered by countable unions of \(C^∞\) surfaces and its projection onto the time axis has dimension zero.
In particular, we show that the free boundary of the Stefan problem in three spatial dimensions will be a smooth surface outside of a set of times of dimension at most 1/2.