Errata to “Smooth convergence away from singular sets”

2020 ◽  
Vol 28 (7) ◽  
pp. 1755-1772
Author(s):  
Sajjad Lakzian ◽  
Christina Sormani
Keyword(s):  
Singular Sets ◽  
Smooth Convergence

scholarly journals Smooth convergence away from singular sets

2013 ◽  
Vol 21 (1) ◽  
pp. 39-104 ◽  
Author(s):  
Sajjad Lakzian ◽  
Christina Sormani
Keyword(s):  
Singular Sets ◽  
Smooth Convergence

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

Analytic Functions with an Irregular Linearly Measurable Set of Singular Points

1952 ◽  
Vol 4 ◽  
pp. 424-435 ◽  
Author(s):  
I. E. Glover
Keyword(s):  
Imaginary Part ◽  
Analytic Functions ◽  
Singular Points ◽  
Definite Integrals ◽  
Singular Sets ◽  
Dense Point ◽  
Point Set ◽  
Dense Point Set ◽  
Nowhere Dense Set ◽  
Dense Set

V. V. Golubev, in his study [6], has constructed, by using definite integrals, various examples of analytic functions having a perfect nowhere-dense set of singular points. These functions were shown to be single-valued with a bounded imaginary part. In attempting to extend his work to the problem of constructing analytic functions having perfect, nowhere-dense singular sets under quite general conditions, he posed the following question: Given an arbitrary, perfect, nowhere-dense point-set E of positive measure in the complex plane, is it possible to construct, by passing a Jordan curve through E and by using definite integrals, an example of a single-valued analytic function, which has E as its singular set, with its imaginary part bounded.

Non-differentiable functionals and singular sets of minima

2005 ◽  
Vol 340 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Jan Kristensen ◽  
Giuseppe Mingione
Keyword(s):  
Singular Sets

Nodal sets and horizontal singular sets of ℍ-harmonic functions on the Heisenberg group

2014 ◽  
Vol 16 (04) ◽  
pp. 1350049 ◽  
Author(s):  
Long Tian ◽  
Xiaoping Yang
Keyword(s):  
Heisenberg Group ◽  
Geometric Structure ◽  
Harmonic Functions ◽  
Nodal Sets ◽  
Singular Sets ◽  
Definition Of

In this paper, we give measure estimates of nodal sets of ℍ-harmonic functions on the Heisenberg group ℍn. We also introduce a definition of horizontal singular sets and show the geometric structure of the horizontal singular sets of ℍ-harmonic functions.

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