Analysis and Probability Seminar

A Functional Analytic Approach to Self-improvement Properties

Simon Bortz (University of Minnesota)

Friday, March 9, 2018 1:30 pm
MONT 414

In 1963, Norman Meyers showed that $W^{1,2}_{loc}$ solutions to divergence form elliptic operators have a gradient in $L^p$ for some $p > 2$. Modern proofs of this fact utilize the Gehring Lemma (1973). This property can be observed in fractional integro-differential equations. In fact, one can show that solutions to these fractional equations improve in differentiability as well; this was recently investigated by Kuusi, Mingione and Sire.

Led by this, my co-authors and I sought to exploit `hidden' non-local structure in parabolic equations to prove that solutions are locally Hölder in time with values in $L^p$ for some $p > 2$. While the `Gehring' approach was effective, we found an alternative functional analytic proof that is quite simple. I will show how to apply this functional analysis approach to parabolic equations in detail, then I will indicate how this method applies to other equations. This is joint work with P. Auscher, M. Egert and O. Saari.