Parabolic Theory as a High-Dimensional Limit of Elliptic Theory

Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a general method for producing parabolic theorems from their elliptic analogues. Recent results for variable-coefficient and non-local operators, joint work with Mariana Smit Vega Garcia, will be discussed.