Parabolic Signorini Problem
The Signorini problem is a prototypical example of so-called variational inequalities, exhibiting free boundaries. Originally stated for an elastic body resting on a rigid surface, mathematically it is similar to many other problems such as boundary heat control problems, problems with semipermeable membranes, modeling of American options governed by jump processes, etc.
In this talk we will discuss the parabolic (time-dependent) version of the problem, including the results on the optimal regularity of the solutions, structure and smoothness of the free boundary. We will also talk about generalizations of a celebrated monotonicity formula of Almgren, which is an important technical tool in our study.