Atlanta, GA | Posted: September 27, 2007
Robert V. Kohn is a world-renowned expert on nonlinear partial differential equations and nonconvex variational problems. He has worked on a range of problems motivated by applications in materials science and, more recently, financial mathematics. He was selected as Plenary Speaker for the 2006 International Congress of Mathematicians (ICM) and for the 2007 International Congress on Industrial and Applied Mathematics (ICIAM).
Energy-driven pattern formation is difficult to define, but easy to recognize. I'll discuss two examples: (a) cross-tie wall patterns in magnetic thin films, and (b) surface-energy-driven coarsening of two-phase mixtures. The two problems are rather different -- the first is static, the second dynamic. But they share certain features: in each case nature forms complex patterns as it attempts to minimize a suitable "free energy". The task of modeling and analyzing such patterns is a rich source of challenges -- many still open -- in the multidimensional calculus of variations.
We usually think of parabolic partial differential equations and first-order Hamilton-Jacobi equations as being quite different. Parabolic equations are linked to random walks, and often arise as steepest-descents; Hamilton-Jacobi equations have characteristics, and often arise from optimal control problems. In truth, these equations are not so different. I will discuss recent work with Sylvia Serfaty, which provides deterministic optimal-control interpretations of many parabolic PDE. In some cases -- for example motion by curvature -- the optimal control viewpoint is very natural, geometric, and easy to understand. In other cases -- for example the linear heat equation -- it seems a bit less natural, and therefore even more surprising.