Seminar

# Parabolic uniform rectifiability and caloric measure I: $A_\infty$ implies parabolic uniform rectifiability of a parabolic Lipschitz graph.

#### Geometric Aspects of Nonlinear Partial Differential Equations

#### 22 November 15:00 - 16:00

#### Kaj Nyström - Uppsala University

We prove that if a parabolic Lipschitz graph domain has the property that its caloric measure is a parabolic $A_\infty$ weight with respect to surface measure, then the function defining the graph has a half-order time derivative in the space of (parabolic) bounded mean oscillation. Equivalently, we prove that the $A_\infty$ property of caloric measure implies that the boundary is parabolic uniformly rectifiable. Consequently, by combining our result with the work of Lewis and Murray we prove that the $L^p$ solvability (for some $p > 1$) of the Dirichlet problem for the heat equation is equivalent to parabolic uniformly rectifiability. This is joint work with S. Bortz, S. Hofmann, and J.M. Martell.

Organizers

Panagiota Daskalopoulos

Columbia University

Alessio Figalli

ETH Zürich

Erik Lindgren

Uppsala University

Henrik Shahgholian

KTH Royal Institute of Technology

Susanna Terracini,

University of Turin