We introduce a new notion of a harmonic measure for a d-dimensional set in Rn with d<n−1, that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.
Bibliographical noteFunding Information:
Svitlana Mayboroda is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089. Guy David is supported in part by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014. Joseph Feneuil is partially supported by the ANR project ?HAB? No. ANR-12-BS01-0013. This work was supported by a public grant as part of the ?Investissement d'avenir? project, reference ANR-11-LABX-0056-LMH, LabEx LMH. Part of this work was completed during S. Mayboroda's visit to Universit? Paris-Sud, Laboratoire de math?matiques, Orsay, and ?cole polytechnique, PMC. We thank the corresponding Departments and Fondation Jacques-Hadamard for support and hospitality.