The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n − 1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.
Bibliographical noteFunding Information:
Received June 18, 2018, accepted January 25, 2019 The first author was partially supported by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014, the European Community Marie Curie grant MANET 607643 and H2020 grant GHAIA 777822, and the Simons Collaborations in MPS grant 601941, GD. The third author was supported by the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089, the NSF RAISE-TAQ grant DMS 1839077, the Simons Fellowship, and the Simons Foundation grant 563916, SM. 1) Corresponding author
© 2019, Springer-Verlag GmbH Germany & The Editorial Office of AMS.
- Dahlberg’s theorem
- Dirichlet solvability
- Elliptic measure in higher codimension
- absolute continuity
- degenerate elliptic operators