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A new elliptic measure on lower dimensional sets. (arXiv:1807.07035v1 [math.AP])
来源于:arXiv
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. 查看全文>>