摘要
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.
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.
基金
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
the Simons Collaborations in MPS grant 601941,GD
supported by the NSF INSPIRE Award DMS1344235
NSF CAREER Award DMS 1220089
the NSF RAISE-TAQ grant DMS 1839077
the Simons Fellowship
the Simons Foundation grant 563916,SM
