A min-max representation of elliptic operators, and applications

Russell Schwab
Michigan State University

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.