Hausdorff Measure Research Articles

Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation

We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation −Δs in the critical case of s=32, which satisfy certain local energy inequalities and whose singular sets have a locally finite two-dimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for s=32. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when s=32, the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentration-compactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.

Geometry of planar curves intersecting many lines at a few points

The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is σ \sigma -finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.

Compactness in Groups of Group-Valued Mappings

We introduce the concepts of extended equimeasurability and extended uniform quasiboundedness in groups of group-valued mappings endowed with a topology that generalizes the topology of convergence in measure. Quantitative characteristics modeled on these concepts allow us to estimate the Hausdorff measure of noncompactness in such a contest. Our results extend and encompass some generalizations of Fréchet–Šmulian and Ascoli–Arzelà compactness criteria found in the literature.

Interior and boundary regularity criteria for the 6D steady Navier-Stokes equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫B1|u(x)|3dx+∫B1|f(x)|qdx or ∫B1|∇u(x)|2dx+∫B1|∇u(x)|2dx(∫B1|u(x)|dx)2+∫B1|f(x)|qdx with q>3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫B1+|u(x)|3dx+∫B1+|f(x)|3dx or ∫B1+|∇u(x)|2dx+∫B1+|f(x)|3dx is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8], Indiana Univ. Math. J., 2012), Dong-Gu ([7], J. Funct. Anal., 2014), and Liu-Wang ([27], J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary.

Solvability of infinite systems of fractional differential equations in the double sequence space $$ 2^c(\triangle )$$

In this work, we first introduce the concept of double sequence space $$2^c(\triangle )$$ . Then, we construct a Hausdorff measure of noncompactness on this space. Furthermore, by employing this measure of noncompactness we discuss the existence of solutions for infinite systems of third-order three-point boundary value problem in the double sequence space $$2^c(\triangle )$$ . Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness

On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness

CONVERGENCE OF DIRICHLET FORMS AND BESOV NORMS ON SCALE IRREGULAR SIERPIŃSKI GASKETS

In this paper, we construct equivalent semi-norms of local and non-local Dirichlet forms on scale irregular Sierpiński gaskets. These fractals are not necessarily self-similar, and have volume doubling Hausdorff measures which are not necessarily Ahlfors regular. We obtain that a sequence of non-local Dirichlet forms converges to a local Dirichlet form, which extends a convergence theorem of Bourgain, Brezis and Mironescu to the scale irregular Sierpiński gaskets for [Formula: see text].

Topological properties of solution sets for Riemann-Liouville fractional nonlocal delay control systems with noncompact semigroups and applications to approximate controllability

This paper investigates the topological properties of the mild solution set for a control system monitored by Riemann-Liouville fractional nonlocal delay evolution equations with noncompact semigroups. We first obtain the Rδ-property of the mild solution set by employing the method of the Hausdorff measure of noncompactness. Then, we apply the obtained result to show that the presented control problem admits a reachable invariant set under nonlinear perturbations. Furthermore, we apply the obtained result to characterize the approximate controllability of the presented control system. Finally, we provide an example to illustrate the application of the abstract results.

Numerical quadrature for singular integrals on fractals

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Gamma subset mathbb {R}^n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Gamma, including in particular the Hausdorff measure mathcal {H}^d restricted to Gamma, where d is the Hausdorff dimension of Gamma. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Gamma are decomposed into sums of integrals over suitable partitions of Gamma into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens.

Eigenvalues of the Birman-Schwinger operator for singular measures: The noncritical case

For operators of the form T=TA,P=A⁎PA with pseudodifferential operator A of negative order −l in a domain in RN, 2l≠N, and a singular measure P, an estimate of eigenvalues is found with an order depending on the dimensional characteristics of the measure P and the coefficient depending on an integral norm of the density of P with respect to the Hausdorff measure of an appropriate dimension. These estimates are used to establish asymptotic formulas for the eigenvalues of T for the case when P is supported on a Lipschitz surface of some codimension and on certain sets of a more complicated structure. In one of applications, a version of the CLR estimate for singular measures is proved.

Some sharp isoperimetric-type inequalities on Riemannian manifolds

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound l for the cut distance and Ricci curvature lower bound (n−1)k, the geodesic ball of radius l in the space form of curvature k has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that C2 isoperimetric domains in standard space forms are balls. We also prove another isoperimetric inequality involving the extrinsic radius of a domain when the curvature of the ambient space is bounded above. We then extend it to an inequality which involves the Hausdorff measure of the cut locus.

Sample distribution theory using Coarea Formula

Let ( Ω , Σ , p ) be a probability measure space and let X : Ω → R k be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on R k and set fX as its density function. Let ϕ : R k → R n be a C 1-map and let us consider the new random variable Y = ϕ ( X ) : Ω → R n . Setting m : = max { rank ( J ϕ ( x ) ) : x ∈ R k } , we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure H m on ϕ ( R k ) which satisfies f Y ( y ) = ∫ ϕ − 1 ( y ) f X ( x ) 1 J m ϕ ( x ) d H k − m ( x ) , for H m − a . e . y ∈ ϕ ( R k ) . Here J m ϕ is the m-dimensional Jacobian of ϕ . When J ϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E ⊆ R k and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and “Student's t” distributions.

BOUNDED VARIATION ON THE SIERPIŃSKI GASKET

Under certain continuity conditions, we estimate upper and lower box dimensions of the graph of a function defined on the Sierpiński gasket. We also give an upper bound for Hausdorff dimension and box dimension of the graph of a function having finite energy. Further, we introduce two sets of definitions of bounded variation for a function defined on the Sierpiński gasket. We show that fractal dimension of the graph of a continuous function of bounded variation is [Formula: see text] We also prove that the class of all bounded variation functions is closed under arithmetic operations. Furthermore, we show that every function of bounded variation is continuous almost everywhere in the sense of [Formula: see text]-dimensional Hausdorff measure.

Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

In the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If Omega is an open domain in mathbb {R}^n with mild topological conditions, we can even characterize the n-1 dimensional uniformly rectifiability of the boundary partial Omega by the A_infty -absolute continuity of the harmonic measure on partial Omega with respect to the surface measure. In low dimension (d<n-1), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if Gamma is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on Gamma is A_infty -absolutely continuous with respect to the d-dimensional Hausdorff measure. In the present article, we use a completely new approach to give an alternative and significantly shorter proof of David and Mayboroda’s result.

Removable singularity of positive mass theorem with continuous metrics

In this paper, we consider asymptotically flat Riemannian manifolds endowed with a continuous metric and the metric is smooth away from a compact subset with certain conditions. If the metric is Lipschitz and the scalar curvature is nonnegative away from a closed subset with \((n-1)\)-dimensional Hausdorff measure zero, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that the manifold is isometric to the Euclidean space. The Hausdorff measure condition is optimal, which confirms a conjecture of Lee in 2013. The proof of the nonnegativity is based on a smoothing argument with some new estimates such as a \(L^1\)-scalar curvature approximation estimates. The proof of the rigidity relies on the RCD theory where we show that the manifold has nonnegative Ricci curvature in RCD sense. The idea involving RCD theory in the proof turns out to be much more suitable in the study of the rigidity for continuous metrics. Furthermore, our result generalizes the result of Lee and LeFloch (Commun Math Phys 339(1):99–120, 2015) from spin to non-spin.

Fractional Gaussian fields on the Sierpiński Gasket and related fractals

We study the regularity of the Gaussian random measures (− Δ)−sW on the Sierpiński gasket where W is a white noise and Δ the Laplacian with respect to the Hausdorff measure. Along the way we prove sharp global Hölder regularity estimates for the fractional Riesz kernels on the gasket which are new and of independent interest.

Continuous maps that preserve Hausdorff measure

We show that the space of homeomorphisms on the unit interval [0,1], equipped with the topology of uniform convergence, contains a dense subspace of functions that preserve the Hausdorff measure of any subset of certain one-dimensional self-similar sets. We extend the results to a class of Cantor dust type self-similar sets in R2.

The inverse Poletsky inequality in one class of mappings

Almost everywhere differentiable mappings possessing the Luzin N-property, the N–1-property on spheres with respect to the (n–1)-dimensional Hausdorff measure, and the zero Lebesgue measure of the image of points where their Jacobian vanishes, have been studied. It has been proved that such mappings satisfy the lower estimate of the Poletsky-type distortion in their domain. In particular, if homeomorphism has an inverse from the Orlicz–Sobolev class, then, provided the Calderón condition for the determining function holds, it satisfies the inverse Poletsky inequality.

Lipschitz mappings, metric differentiability, and factorization through metric trees

Given a Lipschitz map f $f$ from a cube into a metric space, we find several equivalent conditions for f $f$ to have a Lipschitz factorization through a metric tree. As an application, we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if f $f$ is a Lipschitz mapping from an open set in R n $\mathbb {R}^n$ onto a metric space X $X$ , then the topological dimension of X $X$ equals n $n$ if and only if X $X$ has positive n $n$ -dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.

Solvability of infinite system of nonlinear convolution type integral equations in the tempered sequence space mβ(φ,p)

In this paper, we first introduce the concept of tempered sequence space [Formula: see text]. Then, we construct a Hausdorff measure of noncompactness on this sequence space. Furthermore, by employing this measure of noncompactness we discuss the existence of solutions for an infinite system of nonlinear convolution differential equations in the tempered sequence space [Formula: see text]. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.