Finite Hausdorff Research Articles

On removable singularities of harmonic functions on a stratified set

We consider sets removable for bounded harmonic functions on a stratified set with flat interior strata. We establish that relatively closed sets of finite Hausdorff (n-2)-measure are removable for bounded harmonic functions on an n-dimensional stratified set satisfying the “strong sturdiness” condition.

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.

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2 -manifold with boundary) with locally finite Hausdorff 2 -measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2 -sphere with finite Hausdorff 2 -measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

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.

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.

Quasiconformal almost parametrizations of metric surfaces

We look for minimal conditions on a two-dimensional metric surface X of locally finite Hausdorff 2-measure under which X admits an (almost) parametrization with good geometric and analytic properties. Only assuming that X is locally geodesic, we show that Jordan domains in X of finite boundary length admit a quasiconformal almost parametrization. If X satisfies some further conditions, then such an almost parametrization can be upgraded to a geometrically quasiconformal homeomorphism or a quasisymmetric homeomorphism. In particular, we recover Rajala’s recent quasiconformal uniformization theorem in the special case that X is locally geodesic as well as Bonk–Kleiner’s quasisymmetric uniformization theorem. On the way, we establish the existence of Sobolev discs spanning a given Jordan curve in X under nearly minimal assumptions on X and prove the continuity of energy minimizers.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

Coarea inequality for monotone functions on metric surfaces

We study coarea inequalities for metric surfaces — metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure H 2 \mathcal {H}^2 . For monotone Sobolev functions u : X → R u\colon X \to \mathbb {R} , we prove the inequality ∫ R ∗ ∫ u − 1 ( t ) g d H 1 d t ≤ κ ∫ X g ρ d H 2 for every Borel g : X → [ 0 , ∞ ] , \begin{equation*} \int _{ \mathbb {R} }^{*} \int _{ u^{-1}(t) } g \,d\mathcal {H}^{1} \,dt \leq \kappa \int _{ X } g \rho \,d\mathcal {H}^{2} \quad \text {for every Borel $g \colon X \rightarrow \left [0,\infty \right ]$,} \end{equation*} where ρ \rho is any integrable upper gradient of u u . If ρ \rho is locally L 2 L^2 -integrable, we obtain the sharp constant κ = 4 / π \kappa =4/\pi . The monotonicity condition cannot be removed as we give an example of a metric surface X X and a Lipschitz function u : X → R u \colon X \to \mathbb {R} for which the coarea inequality above fails.

Tensorization of quasi-Hilbertian Sobolev spaces

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space X\times Y can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, W^{1,2}(X\times Y)=J^{1,2}(X,Y) , thus settling the tensorization problem for Sobolev spaces in the case p=2 , when X and Y are infinitesimally quasi-Hilbertian , i.e., the Sobolev space W^{1,2} admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces X,Y of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally, for p\in (1,\infty) we obtain the norm-one inclusion \|f\|_{J^{1,p}(X,Y)}\le \|f\|_{W^{1,p}(X\times Y)} and show that the norms agree on the algebraic tensor product W^{1,p}(X)\otimes W^{1,p}(Y)\subset W^{1,p}(X\times Y). When p=2 and X and Y are infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of W^{1,2}(X)\otimes W^{1,2}(Y) in J^{1,2}(X,Y) , thus implying the equality of the spaces. Our approach raises the question of the density of W^{1,p}(X)\otimes W^{1,p}(Y) in J^{1,p}(X,Y) in the general case.

Examples of d-sets with Irregular Projection of Hausdorff Measures

Given positive integers $$\ell <n$$ and a real $$d\in (\ell ,n)$$ , we construct sets $$K\subset \mathbb {R}^n$$ with positive and finite Hausdorff d–measure such that the Radon–Nikodym derivative associated to all projections on $$\ell $$ -dimensional planes, when it exists, is not an $$L^p$$ function for any $$p>1$$ .

Dimension Estimate of the Global Attractor for a 3D Brinkman- Forchheimer Equation

In this paper, we study the dimension estimate of global attractor for a 3D Brinkman-Forchheimer equation. Based on the differentiability of the semigroup with respect to the initial data, we show that the global attractor of strong solution of the 3D Brinkman-Forchheimer equation has finite Hausdorff and fractal dimensions.

Dimension results and local times for superdiffusions on fractals

We consider the Dawson–Watanabe superprocess obtained from a spatial motion with sub-Gaussian transition densities on a metric measure space with finite Hausdorff dimension, and examine the dimensions of the range and the set of times when the support intersects a given set, generalising results of Serlet and Tribe. As intermediate results, we prove existence of local times for the superprocess if the spectral dimension of the spatial motion satisfies ds<4, and prove that (2−ds/2)∧1 is the critical Hölder-continuity exponent in the time variable. Furthermore, we prove a bound on moments of the integrated superprocess, and give uniform upper bounds on the mass the superprocess assigns to small balls, generalising a result of Perkins.

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.

Big Ramsey Spectra of Countable Chains

A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of countable chains with finite big Ramsey spectra (or degrees) one still has to resolve the remaining case of countable scattered chains of finite Hausdorff rank whose finite sums are not bounded.

On the sharp lower bound for duality of modulus

We establish a sharp reciprocity inequality for modulus in compact metric spaces X X with finite Hausdorff measure. In particular, when X X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E E and F F in X X and the modulus of the family of surfaces of finite Hausdorff measure that separate E E and F F . The paper also develops approximation techniques, which may be of independent interest.

On the Hausdorff dimension of the residual set of a packing by smooth curves

Let a planar residual set be a set obtained by removing countably many disjoint topological disks from an open set in the plane. We prove that the residual set of a planar packing by curves that satisfy a certain lower curvature bound has Hausdorff dimension bounded away from 1, quantitatively, depending only on the curvature bound. As a corollary, the residual set of any circle packing has Hausdorff dimension uniformly bounded away from 1. This result generalizes the result of Larman, who obtained the same conclusion for circle packings inside a square. We also show that our theorem is optimal and does not hold in general without lower curvature bounds. In particular, we construct packings by strictly convex, smooth curves whose residual sets have dimension 1. On the other hand, we prove that any packing by strictly convex curves cannot have $\sigma$-finite Hausdorff 1-measure.

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method; Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek; Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

Uniformization of metric surfaces using isothermal coordinates

&#x0D; We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.

The partial regularity for Landau-Lifshitz-Maxwell-Spin diffusion system in three dimensions

In this paper, we consider the model of the Landau-Lifshitz-Maxwell system coupled with spin accumulation. This system consists of the Landau-Lifshitz equation, Maxwell equation describing the magnetic field and a quasi-linear parabolic equation describing the process of spin accumulation. This paper establishes the existence of weak solutions (M,S,H,E) to the coupled system in three dimensions and shows the magnetization vector field M is Hölder continuous except a closed set Σ, which has locally finite 3-dimensional parabolic Hausdorff measure. Finally, we deduce the Hölder continuity of ∇M away from Σ in two special cases.