Title: Lecture 3: Harnack inequalities for isoperimetric surfaces and free boundaries
Speaker: David Jerison (MIT)
Abstract: The behavior of level sets of harmonic functions and eigenfunctions can be subsumed in the more general question of behavior of level surfaces of solutions u to a semilinear elliptic equation, ∆ u = f (u). The most interesting singular limits of these families of level sets are isoperimetric surfaces and free boundaries. In the third lecture, we will discuss Harnack inequalities for such interfaces. These are quantitative forms of the maximum principle that yield global rigidity and local regularity results for the surfaces. We will speculate on how these Harnack inequalities should relate to geometric differential Harnack inequalities.