On the blow-up of GSBV functions under suitable geometric properties of the jump set

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Emanuele Tasso
Keyword(s):  
Hausdorff Dimension ◽  
Sobolev Spaces ◽  
Blow Up ◽  
Geometric Properties ◽  
Open Set ◽  
Geometric Conditions ◽  
Jump Set

AbstractIn this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set {\Omega\subset\mathbb{R}^{n}} and given {p>1}, we study the blow-up of functions {u\in\mathrm{GSBV}(\Omega)}, whose jump sets belong to an appropriate class {\mathcal{J}_{p}} and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to {n-p}. Moreover, we are able to prove the following result which in the case of {W^{1,p}(\Omega)} functions can be stated as follows: whenever {u_{k}} strongly converges to u, then, up to subsequences, {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most {n-p}.

scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

scholarly journals Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms

1989 ◽  
Vol 9 (3) ◽  
pp. 403-425 ◽  
Author(s):  
Lorenzo J. Diaz ◽  
Marcelo Viana
Keyword(s):  
Hausdorff Dimension ◽  
Equilibrium Measures ◽  
Open Set ◽  
Limit Capacity ◽  
Nonwandering Set

AbstractWe consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

scholarly journals On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Besov Space ◽  
Sobolev Spaces ◽  
Wavelet Coefficients ◽  
Multifractal Formalism ◽  
Spectrum Of A Function ◽  
Set Of Points

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

scholarly journals GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

2008 ◽  
Vol 05 (05) ◽  
pp. 699-754 ◽  
Author(s):  
JOSEF JANYŠKA ◽  
MARCO MODUGNO
Keyword(s):  
Electromagnetic Field ◽  
Phase Space ◽  
Relativistic Particle ◽  
Jet Space ◽  
Geometric Structures ◽  
Geometric Properties ◽  
Geometric Conditions ◽  
General Relativistic ◽  
Relativistic Phase

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

scholarly journals A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

2009 ◽  
Vol 104 (1) ◽  
pp. 132 ◽  
Author(s):  
Mihai Mihailescu ◽  
Vicentiu Radulescu
Keyword(s):  
Continuous Function ◽  
Eigenvalue Problem ◽  
Continuous Spectrum ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Open Set ◽  
Quasilinear Problem

We study the nonlinear eigenvalue problem $-(\mathrm{div} (a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in ${\mathsf R}^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

2020 ◽  
Vol 17 (03) ◽  
pp. 501-557
Author(s):  
Hao Li ◽  
Yachun Li
Keyword(s):  
Blow Up ◽  
Initial Density ◽  
Local Existence ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Hydrodynamic Equations ◽  
Open Set ◽  
The Cauchy Problem ◽  
Local Strong Solutions ◽  
Blow Up Criterion

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

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