This colloquium doubles as the opening talk for the Workshop on Harmonic Analysis and Partial Differential Equations.
Please note, that this colloquium is taking place at Lawrence Hall (lecture room 120) and the coffee and cookies will be taking place after this talk at 4:30 p.m. instead of before.
120 Lawrence Hall
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For an open set Ω ⊂ Rd with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplace’s equation, with boundary data in Lp for some p < ∞, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure on ∂Ω. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to “surface measure” on the parabolic Ahlfors regular (lateral) boundary Σ, implies parabolic uniform rectifiability of Σ. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem.
This is joint work with S. Bortz, J. M. Martell and K. Nystr¨om.