New perspectives on evolutionary free boundary problems

The purpose of this project is to develop new perspectives on evolutionary Free Boundary Problems (FBPs), i.e. free boundary problems in which the free boundary is moving in both space and time, based on techniques from harmonic analysis and geometric measure theory.

A central problem in the context of FBPs, in theory as well as in applications, is the  difficult task of understanding the geometry of the free boundary. We intend to 1.

Prove that Parabolic Uniform Rectifiability (PUR) provides in many respects the correct geometrical framework for evolutionary one-phase and two-phase FBPs for second order parabolic equations exemplified by the heat equation; 2. Advance the understanding of PUR by establishing equivalent analytic and geometric characterizations thereof; 3.

Establish the robustness of the techniques developed by addressing similar problems for non-linear parabolic equations exemplified by the evolutionary p-Laplace equation and for strongly degenerate parabolic operators exemplified by the Kolmogorov-Fokker-Planck equation.The project will in its parts be carried out with international collaborators,  postdocs and Phd-students.Evolutionary FBPs serve as models for many phenomena studied in the natural sciences, engineering and finance.

Despite this the fundamental mathematical analysis of these models is in many respects lacking and in this project we intend to reshape the field  by making progress on fundamental problems in parabolic potential theory.