English
 
User Manual Privacy Policy Disclaimer Contact us
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT
  Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Kolasinski, S. (2015). Geometric Sobolev-like embedding using high-dimensional Menger-like curvature. Transactions of the American Mathematical Society, 367(2), 775-811. doi:10.1090/S0002-9947-2014-05989-8.

Item is

Basic

show hide
Item Permalink: http://hdl.handle.net/11858/00-001M-0000-000E-EB2D-5 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0024-806C-6
Genre: Journal Article

Files

show Files
hide Files
:
1205.4112 (Preprint), 488KB
Name:
1205.4112
Description:
File downloaded from arXiv at 2013-03-26 12:05
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
Technical Metadata:
Copyright Date:
-
Copyright Info:
-
:
S0002-9947-2014-05989-8.pdf (Any fulltext), 512KB
Name:
S0002-9947-2014-05989-8.pdf
Description:
-
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
Technical Metadata:
Copyright Date:
-
Copyright Info:
-
License:
-

Locators

show

Creators

show
hide
 Creators:
Kolasinski, Slawomir1, Author              
Affiliations:
1Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1753352              

Content

show
hide
Free keywords: Mathematics, Functional Analysis, math.FA,
 Abstract: We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2 and an m-dimensional subset S of R^n we also introduce global versions of this discrete curvature, by taking supremum with respect to m+2-l points on S. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on S. Next, we prove that if S is compact and m-Ahlfors regular and if p is greater than ml, then the P. Jones' \beta-numbers of S must decay as r^t with r \to 0 for some t in (0,1). If S is an immersed C^1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C^{1,t} manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent t to the optimal one a = 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem. Moreover, we obtain a qualitative control over the local graph representations of S only in terms of the energy.

Details

show
hide
Language(s):
 Dates: 2012-05-182013-03-0820142015
 Publication Status: Published in print
 Pages: I removed Example 3.11, which was wrong in the sense that the \beta-numbers for this set do not decay as r^2
 Publishing info: -
 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 1205.4112
DOI: 10.1090/S0002-9947-2014-05989-8
 Degree: -

Event

show

Legal Case

show

Project information

show

Source 1

show
hide
Title: Transactions of the American Mathematical Society
  Other : Trans. Amer. Math. Soc.
Source Genre: Journal
 Creator(s):
Affiliations:
Publ. Info: Providence, R.I. : American Mathematical Society
Pages: - Volume / Issue: 367 (2) Sequence Number: - Start / End Page: 775 - 811 Identifier: ISSN: 0002-9947
CoNE: /journals/resource/954925218003