Title:  Parabolic Lipschitz Domains and Caloric Measure

Abstract: Since the pioneering work of Dahlberg, the study of quantitative “L^p” solvability of boundary value problems for elliptic and parabolic operators in non-smooth domains have been of considerable interest. (So much so that I won’t attempt to put sufficient history in this abstract!) Dahlberg’s fundamental contribution to the area was to show that in a Lipschitz domain the density of harmonic measure exists and it satisfies a certain scale invariant L^2 estimate (reverse Hölder inequality). This in turn gives L^2 control of a certain conical/non-tangential maximal function by the L^2 norm of the data, for the solution to the Dirichlet problem (harmonic function with prescribed boundary values).

For quite some time, less was known about the corresponding parabolic problem, that is, the Dirichlet problem for the heat equation with prescribed lateral boundary data. It was conjectured that if the lateral boundary is given by a Lip(1,1/2) graph (parabolic Lipschitz) then Dahlberg’s theorem should hold for the caloric measure. Kaufmann and Wu showed this to be false; however, it was shown by Lewis and Murray that if one imposes a mild additional (fractional) time-regularity on the graph, then the L^p Dirichlet problem is solvable. With Hofmann, Martell and Nyström, I have shown that this additional regularity is, in fact, necessary. This resolves a problem open for 30 years.