Absolute Continuity of the Robin Harmonic Measure On Rough Domains

The question of asbolute continuity, with respect to the reference measure, of the harmonic measure on a domain with rough boundary has been the object of many important results. Here we ask about the similar question, but where the Dirichlet condition u=0

on the boundary is replaced by the softer Robin condition ∂u∂n+au=0 (where, say a greater than 0 is a constant).

We describe the connection with the lung and joint work with S. Decio, M. Engelstein, M. Filoche, S. Mayboroda, M. Michetti.