Caloric Measure and Parabolic Uniform Rectifiability

Abstract: In the late 70's Dahlberg showed that harmonic measure and surface measure are mutually absolutely continuous in Lipschitz domains in $\mathbb{R}^d$ (this was a long standing conjecture). In fact, he showed a stronger quantitative version of mutual absolute continuity , $A_\infty$, which is equivalent to certain $L^p$ estimates on solutions. It was conjectured by Hunt that the same is true in the parabolic setting, that is, for parabolic Lipschitz graph domains; however, this turned out to be false as a counterexample was produced by Kaufman and Wu. On the other hand, it was later shown by Lewis and Murray that if the graphs had a little more time-regularity then Dahlberg's theorem holds.

Together with my co-authors, we have shown the work of Lewis and Murray is sharp. In particular, if a domain is given by the region above a parabolic Lipschitz graph the $A_\infty$ property of caloric measure is equivalent to this extra time regularity. These `regular' parabolic Lipschitz graphs are the prototypical parabolic uniformly rectifiable (P-UR) sets and this project is part of a larger program to characterize P-UR sets by properties of caloric functions/measure.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php