Let $ \Omega \subset \mathbb{R}^{n+1}$, $ n\geq 2$, be a 1sided NTA domain (also known as a uniform domain), i.e., a domain which satisfies interior corkscrew and Harnack chain conditions, and assume that $ \partial \Omega $ is $ n$dimensional AhlforsDavid regular. We characterize the rectifiability of $ \partial \Omega $ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $ \partial \Omega $ can be covered $ \mathcal {H}^n$a.e. by a countable union of portions of boundaries of bounded chordarc subdomains of $ \Omega $ and to the fact that $ \partial \Omega $ possesses exterior corkscrew points in a qualitative way $ \mathcal {H}^n$a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.
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Free boundary regularity in the triple membrane problem
We investigate the regularity of the free boundaries in the three elastic membranes
problem.
We show that the two free boundaries corresponding to the coincidence regions between
consecutive membranes are C1,loghypersurfaces near a regular intersection point. We also
study two types of singular intersections. The first type of singular points are locally covered
by a C1,alphahypersurface. The second type of singular points stratify and each stratum is
locally covered by a C1manifold.
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 Award ID(s):
 1800645
 NSFPAR ID:
 10407546
 Date Published:
 Journal Name:
 Ars inveniendi analytica
 Volume:
 2021
 Issue:
 3
 ISSN:
 27698505
 Page Range / eLocation ID:
 149
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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