scholarly journals Dimension estimates for the boundary of planar Sobolev extension domains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Danka Lučić ◽  
Tapio Rajala ◽  
Jyrki Takanen
Keyword(s):  
Upper Bound ◽  
Simply Connected

Abstract We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev W 1 , p {W^{1,p}} -extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.

scholarly journals On Conformal Radii of Non-Overlapping Simply Connected Domains

2018 ◽  
Vol 11 ◽  
pp. 1-7
Author(s):  
Yaroslav V. Zabolotnyi ◽  
Iryna Denega
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Conformal Radius ◽  
Simply Connected ◽  
Disjoint Domains ◽  
Simply Connected Domains

The paper deals with the following open problem stated by V.N. Dubinin. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\subset \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains. For all values of the parameter $\gamma\in (0, n]$ find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the conformal radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geqslant2$ the problem was solved by V.N. Dubinin. In the paper the problem is solved for $\gamma\in (0, \sqrt{n}\,]$ and $n\geqslant2$ for simply connected domains.The paper deals with the following open problem stated by V.N. Dubinin. Let a0 = 0, ιa1ι =...= ιanι = 1, ak ∈ Bk ⊂ , where B0, ..., Bn are disjoint domains. For all values of the parameter γ∈ (0; n] find the exact upper bound nfor rγ(B0; 0) ∏ r(Bk; ak), where r(Bk; ak) is the conformal radius of Bk with respect to ak. For γ = 1 k=1 and n ≥ 2 the problem was solved by V.N. Dubinin. In the paper the problem is solved for γ ∈ (0; √n ] and n ≥ 2 for simply connected domains.

scholarly journals Upper Bounds for the Essential Dimension of E7

2013 ◽  
Vol 56 (4) ◽  
pp. 795-800 ◽  
Author(s):  
Mark L. MacDonald
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Connected Group ◽  
Simply Connected

Abstract.This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E7 over a field of characteristic not 2 or 3. In particular, ed(E7) ≤ 29, and ed(E7, 2) ≤ 27.

scholarly journals Estimation of Riemannian Barycentres

2004 ◽  
Vol 7 ◽  
pp. 193-200 ◽  
Author(s):  
Huiling Le
Keyword(s):  
Riemannian Manifolds ◽  
Upper Bound ◽  
Iterative Procedure ◽  
Jacobi Field ◽  
Probability Measures ◽  
Shape Spaces ◽  
Sectional Curvatures ◽  
Simply Connected

AbstractUsing Jacobi field arguments, this paper describes an iterative procedure for finding the Riemannian barycentres of a class of probability measures on complete, simply connected Riemannian manifolds with a finite upper bound on their sectional curvatures. This, in particular, generalises an earlier result of the author's (‘Locating Fréchet means with application to shape spaces’, Adv. Appl. Probab. 33 (2001) 324-338).

scholarly journals Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

2021 ◽  
Author(s):  
Bruno Colbois ◽  
Alessandro Savo
Keyword(s):  
Magnetic Field ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Upper Bounds ◽  
Zero Magnetic Field ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Simply Connected ◽  
Lowest Eigenvalue

AbstractWe obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain $$\Omega $$ Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $$\epsilon $$ ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

Sign in / Sign up

Export Citation Format

Share Document