Speaker: Ivan Blanc

Abstract: Many books on PDEs begin their discussion of elliptic PDEs by focusing on harmonic functions and the Laplacian.  After all, in addition to all of the applied mathematics and physics that involves Laplace's equation, there are immediate and strong connections to complex analysis and probability.  Also, the development of the theory for Laplace's Equation is greatly accelerated by immediate access to the mean value theorem which states that the mean value of a harmonic function on a ball will give you the value of that function at the center of that ball.  The usual proof of this theorem relies heavily on the symmetry and smoothness properties of the Laplace operator, however, in the Fermi lectures on the obstacle problem, Caffarelli outlined a way to prove an analogous theorem for arbitrary divergence form uniformly elliptic operators by looking at the noncontact set of an obstacle problem.