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