A New Elliptic Measure on Lower Dimensional Sets

Guy David, Joseph Feneuil, Svitlana Mayboroda

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)876-902
Number of pages27
JournalActa Mathematica Sinica, English Series
Volume35
Issue number6
DOIs
StatePublished - Jun 1 2019

Fingerprint

Geometric Measure Theory
Degenerate Elliptic Operators
Absolute Continuity
Scale Invariant
Codimension
Theorem
Class

Keywords

  • 28A75
  • 28A78
  • 31B05
  • 42B20
  • 42B25
  • 42B37
  • Dahlberg’s theorem
  • Dirichlet solvability
  • Elliptic measure in higher codimension
  • absolute continuity
  • degenerate elliptic operators

Cite this

A New Elliptic Measure on Lower Dimensional Sets. / David, Guy; Feneuil, Joseph; Mayboroda, Svitlana.

In: Acta Mathematica Sinica, English Series, Vol. 35, No. 6, 01.06.2019, p. 876-902.

Research output: Contribution to journalArticle

David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana. / A New Elliptic Measure on Lower Dimensional Sets. In: Acta Mathematica Sinica, English Series. 2019 ; Vol. 35, No. 6. pp. 876-902.
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