Abstract

Over the past decade the lower-dimensional, or Signorini, problem has received a great deal of attention, especially after the 2004 breakthrough result of Athanasopoulos and Caffarelli on the optimal C^{1,1/2} smoothness of the solution up to the thin manifold. However, until recently, there has been no significant progress on the parabolic counterpart of such optimal regularity and on the regularity of the free boundary. In this lecture I will discuss recent joint work with D.Danielli, A. Petrosyan and T. To in which we give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren’s monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.