Interaction of a free boundary with a planar diffusion.

The situation is the following: a horizontal plane, having a diffusion of its own, exchanges with the lower half space. There, a  reaction-diffusion process, modeled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane.

The origin of this problem is a two-dimensional reaction diffusion model  proposed some time ago (collaboration with H. Berestycki and L. Rossi) to model  biological invasions. Some counter-intuitive numerical simulations had been  explained (collaboration with L. Caffarelli) by transforming the model into a free boundary interacting with a line, and a careful study of the free boundary. At  this occasion, it was noticed that the free boundary very much like that of the obstacle problem.

The goal of this talk is to explain how the analogy with the obstacle  problem can be pushed further in higher space dimensions.

Joint work with L. Caffarelli and I. Tomasetti.