Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0= \nabla u\left(X) \right\} , has finite ( d − 2 ) \left(d2) dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} Dini domains is sharp, by constructing a large class of nonDini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d2} measures.
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Rectifiability of planes and Alberti representations
We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let X be a pointwise doubling metric measure space. Let U be a Borel subset on which the blowups of X are topological planes. We show that U can admit at most 2 independent Alberti representations. Furthermore, if U admits 2 Alberti representations, then the restriction of the measure to U is 2rectifiable. This is a partial answer to the case n=2 of a question of the second author and Schioppa.
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 Award ID(s):
 1758709
 NSFPAR ID:
 10108544
 Date Published:
 Journal Name:
 Annali della Scuola normale superiore di Pisa. Classe di scienze
 Volume:
 XIX
 Issue:
 2
 ISSN:
 0391173X
 Page Range / eLocation ID:
 723756
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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