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.
Panagiota Daskalopoulos
Columbia University
Alessio Figalli
ETH Zürich
Erik Lindgren
Uppsala University
Henrik Shahgholian
KTH Royal Institute of Technology
Susanna Terracini,
University of Turin


Erik Lindgren

Henrik Shahgholian


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