scholarly journals Regularity of area minimizing currents mod p

2020 ◽  
Vol 30 (5) ◽  
pp. 1224-1336
Author(s):  
Camillo De Lellis ◽  
Jonas Hirsch ◽  
Andrea Marchese ◽  
Salvatore Stuvard
Keyword(s):  
Hausdorff Dimension ◽  
Partial Regularity ◽  
Singular Set ◽  
Regularity Theorem ◽  
Locally Finite ◽  
Dimensional Measure ◽  
Area Minimizing ◽  
Mod P

AbstractWe establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

A partial regularity theorem for harmonic maps at a free boundary

1989 ◽  
Vol 2 (4) ◽  
pp. 299-343 ◽  
Author(s):  
Frank Duzaar ◽  
Klaus Steffen
Keyword(s):  
Free Boundary ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Regularity Theorem

scholarly journals Singular Sets of Some Kleinian Groups (II)

1967 ◽  
Vol 29 ◽  
pp. 145-162 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Singular Set ◽  
Fundamental Domains ◽  
Dimensional Measure ◽  
Singular Sets ◽  
Discontinuous Groups ◽  
Positive Capacity ◽  
Automorphic Functions

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

scholarly journals On the regularity of timelike extremal surfaces

2014 ◽  
Vol 17 (01) ◽  
pp. 1450048 ◽  
Author(s):  
Robert L. Jerrard ◽  
Matteo Novaga ◽  
Giandomenico Orlandi
Keyword(s):  
Hausdorff Dimension ◽  
Minkowski Space ◽  
Specific Form ◽  
Singular Set ◽  
Natural Topology ◽  
One Dimensional ◽  
Class C

We study a class of timelike weakly extremal surfaces in flat Minkowski space ℝ1+n, characterized by the fact that they admit a C1 parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class Ck, then the surface is regularly immersed away from a closed singular set of Euclidean Hausdorff dimension at most 1 + 1/k, and that this bound is sharp. We also show that, generically with respect to a natural topology, the singular set of a timelike weakly extremal cylinder in ℝ1+n is one-dimensional if n = 2, and it is empty if n ≥ 4. For n = 3, timelike weakly extremal surfaces exhibit an intermediate behavior.

scholarly journals Bowen’s formula for meromorphic functions

2011 ◽  
Vol 32 (4) ◽  
pp. 1165-1189 ◽  
Author(s):  
KRZYSZTOF BARAŃSKI ◽  
BOGUSŁAWA KARPIŃSKA ◽  
ANNA ZDUNIK
Keyword(s):  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Topological Pressure ◽  
Singular Set ◽  
Dimension Zero ◽  
Common Value ◽  
Bounded Set ◽  
The Common

AbstractLet f be an arbitrary transcendental entire or meromorphic function in the class 𝒮 (i.e. with finitely many singularities). We show that the topological pressure P(f,t) for t>0 can be defined as the common value of the pressures P(f,t,z) for all z∈ℂ up to a set of Hausdorff dimension zero. Moreover, we prove that P(f,t) equals the supremum of the pressures of f∣X over all invariant hyperbolic subsets X of the Julia set, and we prove Bowen’s formula for f, i.e. we show that the Hausdorff dimension of the radial Julia set of f is equal to the infimum of the set of t, for which P(f,t) is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions f in the class ℬ (i.e. with a bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.

UNIVERSAL 2-BRIDGE KNOT AND LINK ORBIFOLDS

1993 ◽  
Vol 02 (02) ◽  
pp. 141-148 ◽  
Author(s):  
HUGH M. HILDEN ◽  
MARIA TERESA LOZANO ◽  
JOSÉ MARIA MONTESINOS-AMILIBIA
Keyword(s):  
Fundamental Group ◽  
Finite Index ◽  
Discrete Group ◽  
Prime Numbers ◽  
Arithmetic Group ◽  
Singular Set ◽  
Mod P

Let (p/q, n) be the orbifold with cyclic isotropy of order n and with singular set the 2-bridge knot or link p/q where p and q are relatively prime numbers, q is odd, q is less than p, and q is not congruent to ±1 mod p (i.e. p/q is any non toroidal 2-bridge knot or link). We show that the orbifold fundamental group π1(p/q, n) is universal for n any multiple of 12. This means that if Γ is any such group, it can be thought of as a discrete group of hyperbolic isometries of hyperbolic 3-space ℍ3, and then, given any closed, oriented 3-manifold M, there exists a subgroup of finite index G of Γ such that M is homeomorphic to G\ℍ3. Since we have shown elsewhere that the group π1(5/3, 12) is an arithmetic group, it follows that there exists an orbifold, namely (5/3, 12), whose singular set is a knot, the figure eight, and whose fundamental group is both arithmetic and universal.

scholarly journals A note on the Hausdorff dimension of the singular set for minimizers of the Mumford–Shah energy

2014 ◽  
Vol 7 (4) ◽  
Author(s):  
Camillo De Lellis ◽  
Matteo Focardi ◽  
Berardo Ruffini
Keyword(s):  
Hausdorff Dimension ◽  
Singular Set

scholarly journals Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation

Annals of PDE ◽  
2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Maria Colombo ◽  
Silja Haffter
Keyword(s):  
Weak Solution ◽  
Hausdorff Dimension ◽  
Fractional Laplacian ◽  
Compact Set ◽  
Singular Set ◽  
Suitable Weak Solution

AbstractWe consider the SQG equation with dissipation given by a fractional Laplacian of order $$\alpha <\frac{1}{2}$$ α < 1 2 . We introduce a notion of suitable weak solution, which exists for every $$L^2$$ L 2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $$\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) $$ 1 2 α 1 + α α ( 1 - 2 α ) + 2 .

scholarly journals A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

2019 ◽  
Vol 21 (06) ◽  
pp. 1950026 ◽  
Author(s):  
Sergio Conti ◽  
Matteo Focardi ◽  
Flaviana Iurlano
Keyword(s):  
Hausdorff Dimension ◽  
Constitutive Relations ◽  
Type Systems ◽  
Singular Set ◽  
Dimension Formula ◽  
Linear Framework ◽  
Autonomous Case ◽  
Strong Formulation ◽  
Nonlinear Constitutive Relations ◽  
Full Regularity

We prove partial regularity for minimizers to elasticity type energies with [Formula: see text]-growth, [Formula: see text], in a geometrically linear framework in dimension [Formula: see text]. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than [Formula: see text], and actually [Formula: see text] in the autonomous case (full regularity is well-known in dimension [Formula: see text]). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions [Formula: see text] and [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document