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 p...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.展开更多
基金partially supported by the ANR,programme blanc GEOMETRYA ANR-12-BS01-0014the European Community Marie Curie grant MANET 607643 and H2020 grant GHAIA 777822+5 种基金the Simons Collaborations in MPS grant 601941,GDsupported by the NSF INSPIRE Award DMS1344235NSF CAREER Award DMS 1220089the NSF RAISE-TAQ grant DMS 1839077the Simons Fellowshipthe Simons Foundation grant 563916,SM
文摘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.
