The singular set in the obstacle problem

The obstacle problem arises in several important physical models. We will present some recent work in collaboration with A. Figalli and X. Ros-Oton on the structure of the singular set for this problem.

We will start introducing some rather recent tools for the analysis of singularities in the obstacle problem, which are complementary to the classical theory of Caffarelli. These tools exploit a useful connection between singularities of the obstacle problem and solutions of the so-called thin obstacle problem.

With careful enough analysis, we are able to achieve a precise understanding of the behavior of solutions near "generic" singularities.

In particular we prove that the free boundary is generically smooth in dimensions 3 and 4, while in higher dimensions the singular set has, generically, co-dimension 3 inside the free boundary.