Hausdorff Measure Research Articles

Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries

AbstractWhile there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber–Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary (including degenerate singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

Equilibrium measures for two-sided shift spaces via dimension theory

Abstract Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

Mappings of finite distortion on metric surfaces

AbstractWe investigate basic properties of mappings of finite distortion$$f:X \rightarrow \mathbb {R}^2$$ f : X → R 2 , where X is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite 2-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant $$f:X \rightarrow \mathbb {R}^2$$ f : X → R 2 with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if f is moreover injective then $$f^{-1}$$ f - 1 is a Sobolev map.

Compact operators on the Motzkin sequence space $c_0(\mathcal{M})$

The concept of non-compactness measure is extremely beneficial for functional analysis in theories, such as fixed point and operator equations. Apart from these, the Hausdorff measure of non-compactness also has some applications in the theory of sequence spaces which is an interesting topic of functional analysis. One of these applications is to obtain necessary and sufficient conditions for the matrix operators between Banach coordinate (BK) spaces to be compact. In line with these explanations, in this study, the necessary and sufficient conditions for a matrix operator to be compact from the Motzkin sequence space $c_0(\mathcal{M})$ to the sequence space $\mu\in\{\ell_{\infty},c,c_0,\ell_1\}$ are presented by using Hausdorff measure of non-compactness.

Multifractal Characteristics of Gain Structures: A Universal Law of Polycrystalline Strain-Hardening Behaviors

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the strain-hardening exponent by experimental research and fractal geometry analysis. From a geometric perspective, the correlation dimension of grains is consistent with that of the equivalent plastic strain field, which represents the correlation dimension of the material manifold. According to the definition of the Hausdorff measure and Ludwik constitutive model, the strain-hardening exponent represents the exponent derived from the Dth root of the measure relationship. This universal law indicates that the strain-hardening behaviors are fractal geometrized and that the strain-hardening exponent represents a geometrical parameter reflecting the multifractal characteristics of grain structures. This conclusion can enhance the comprehension of the relationship between microstructure and mechanical properties of materials and highlights the importance of designing materials with non-uniform grain distributions to achieve desired hardening properties.

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

Ahlfors regularity of continua that minimize maxitive set functions

The primary objective of this paper is to establish the Ahlfors regularity of minimizers of set functions that satisfy a suitable maxitive condition on disjoint unions of sets. Our analysis focuses on minimizers within continua of the plane with finite 1-dimensional Hausdorff measure. Through quantitative estimates, we prove that the length of a minimizer inside the ball centered at one of its points is comparable to the radius of the ball. By operating within an abstract framework, we are able to encompass a diverse range of entities, including the inradius of a set, the maximum of the torsion function, and spectral functionals defined in terms of the eigenvalues of elliptic operators. These entities are of interest for several applications, such as structural engineering, urban planning, and quantum mechanics.

Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line

For any N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 3$$\\end{document}, we study invariant measures of the dynamics of N hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all N hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable N-body scattering maps which generate a billiard dynamics on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} admitting a canonical weighted Hausdorff measure on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} (that we term the Liouville measure onM\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document}) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linearN-body scattering map yields a billiard dynamics which admits the Liouville measure on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {M}$$\\end{document} as an invariant measure.

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

The set of intersections of several independent Brownian motions on Carnot group

In this paper the existence of intersections for functions of several Brownian motions on the Carnot group is studied. A condition is presented for the existence of such intersections with Probability 1, which is in the form of the asymptotics of a measure on a specific family of small balls. The measure is arbitrary but can be chosen as a surface measure on the manifold related to intersections, and the balls are constructed using the distances related to the processes. Additionally, if the same condition holds in a weaker form, it is shown that there is a Hausdorff measure, such that the value of this Hausdorff measure on the set of intersection points is finite with Probability 1.

Almost sharp lower bound for the nodal volume of harmonic functions

AbstractThis paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove where the doubling index is a notion of growth defined by This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are the notion of stable growth, and a multiscale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant imuprovement over the previous best‐known bound , which implied Nadirashvili's conjecture.

Triebel–Lizorkin capacity and Hausdorff measure in metric spaces with chain condition

Triebel–Lizorkin capacity and Hausdorff measure in metric spaces with chain condition

A beam that can only bend on the Cantor set

In this work, we address the following question: Is it possible for a one‐dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by , each one only able to bend on the set associated with the ‐th step in the construction of the Cantor set and compute the ‐limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension , which is the dimension of the Cantor set.

Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

We explore a new kind of second-order impulsive stochastic differential system with delays. By using the Hausdorff measure of noncompactness and M o ¨ nch fixed point theorem, we investigated the existence of mild solution for the considered systems. Further, we proved the existence of optimal control of the proposed system by using minimizing sequence, Lagrange functional, and Marzur lemma. An example is provided to validate the obtained results.

Differential equations with maximal monotone operators

The paper deals with multivalued differential equations in abstract spaces. Nonlocal conditions are assumed. The model includes an m-dissipative multioperator which generates an equicontinuous, not necessarily compact, semigroup. The regularity of the nonlinear term also depends on the Hausdorff measure of noncompactness. The existence of integral solutions is discussed, with a topological index argument. A transversality condition is required. The results are applied to a partial differential inclusion in a bounded domain in Rn with nonlocal integral conditions. The model also includes an m-dissipative but not necessarily compact semigroup generated by a suitable subdifferential operator.

The Duffin–Schaeffer conjecture for systems of linear forms

AbstractWe extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no ‐by‐ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When , this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where and , in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the cases of that. Catlin's classical conjecture, where , follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where and , follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.

The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a “regular” part and a “singular” part, as Alt and Caffarelli have shown in their pioneer work (Alt and Caffarelli, 1981 [1]) that regular points are C1,γ in two-dimensions. Later, Weiss (1999) [34] first realized that in higher dimensions a critical dimension d⁎ exists so that the singularities of the free boundary can only occur when d⩾d⁎.In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a C1,γ function for some γ∈(0,1). In addition, there exists a critical dimension d⁎ so that the singular set is empty if d<d⁎, discrete if d=d⁎ and of locally finite Hd−d⁎ Hausdorff measure if d>d⁎. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.

A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness

Abstract In this article, we introduce Fibo-Pascal sequence spaces P p F {P_{p}^{F}} , 0 &lt; p &lt; ∞ {0&lt;p&lt;\infty} , and P ∞ F {P_{\infty}^{F}} through the utilization of the Fibo-Pascal matrix P F {P^{F}} . We establish that both P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} are BK-spaces, enjoying a linear isomorphism with the classical spaces ℓ p {\ell_{p}} and ℓ ∞ {\ell_{\infty}} , respectively. Further contributing to the depth of our investigation, we proceed to derive the Schauder basis of the space P p F {P_{p}^{F}} , alongside an exhaustive computation of the α-, β-, and γ-duals for both spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . Additionally, we undertake the task of characterizing certain classes of matrix mappings pertaining to the spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . The final section of this study is dedicated to the meticulous characterization of compact operators acting in the spaces P 1 F {P_{1}^{F}} , P p F {P_{p}^{F}} , and P ∞ F {P_{\infty}^{F}} by using Hausdorff measure of non-compactness.

The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint

Abstract The starting point of this paper is the study of the asymptotic behavior, as p → ∞ {p\to\infty} , of the following minimization problem: min ⁡ { 1 p ⁢ ∫ Ω | ∇ ⁡ v | p + 1 2 ⁢ ∫ Ω ( v - f ) 2 , v ∈ W 1 , p ⁢ ( Ω ) } . \min\biggl{\{}\frac{1}{p}\int_{\Omega}|\nabla v|^{p}+\frac{1}{2}\int_{\Omega}(% v-f)^{2},\quad v\in W^{1,p}(\Omega)\biggr{\}}. We show that the limit problem provides the best approximation, in the L 2 {L^{2}} -norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the ( N - 1 ) {(N-1)} -Hausdorff measure of the jump set of the function.