Russell Schwab (Michigan State University)

TITLE:  A min-max representation of elliptic operators, and applications

ABSTRACT:  We call operators that enjoy the global comparison property `elliptic'' operators.  This means that the operator preserves
ordering between any two functions in its domain, whose graphs are ordered and that agree at a point-- i.e. the operator evaluated at
this location will have the same ordering.  This is a generalization of the fact that we teach to calculus students that at the point of a
local maximum, any $C^2$ function must satisfy $f''(x_0)\leq 0$.  It turns out that not only does this property serve as a defining feature
for many nonlinear partial differential and integro-differential equations, but furthermore, we will present a recent result that shows
the global comparison property implies such an operator must have a familiar form that is common to nonlinear elliptic equations.  Time
permitting, we will elaborate on what this characterization may mean for the interplay between integro-differential equations and
(nonlinear) Dirichlet-to-Neumann mappings and free boundary problems like the Hele-Shaw flow.