arXiv subjects

Sari Rogovin

Publications and source records attributed to Sari Rogovin.

5 recordsLinked to original sources

arXiv2026

Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincar\'e inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincar\'e inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain.

math.AP↗

arXiv2025

Sharp conditions for preserving uniformity, doubling measure and Poincar\'e inequality under sphericalization

We study sphericalization, which is a mapping that conformally deforms the metric and the measure of an unbounded metric measure space so that the deformed space is bounded. The goal of this paper is to study sharp conditions on the deforming density function under which the sphericalization preserves uniformity of the space, the doubling property of the measure and the support of a Poincar\'e inequality. We also provide examples that demonstrate the sharpness of our conditions.

math.MG↗

arXiv2024

Preserving Besov (fractional Sobolev) energies under sphericalization and flattening

We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point for large radii and equipped with a doubling measure, we make a more specific construction based on the measure and equip it with a weighted measure. This mapping is then shown to preserve the doubling property of the measure and the Besov (fractional Sobolev) energy. The corresponding results for flattening of bounded complete metric spaces are also obtained. Finally, it is shown that for the composition of a sphericalization with a flattening, or vice versa, the obtained space is biLipschitz equivalent with the original space and the resulting measure is comparable to the original measure.

math.FA↗

arXiv2022

Some remarks on the Gehring-Hayman theorem

In this paper we provide new characterizations of the Gehring-Hayman theorem from the point of view of Gromov boundary and uniformity. We also determine the critical exponents for the uniformized space to be a uniform space in the case of the hyperbolic spaces, the model spaces $\mathbb{M}^κ_n$ of the sectional curvature $κ<0$ with the dimension $n \geq 2$ and hyperbolic fillings.

math.MG↗

arXiv2017

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let $X$ be a metric measure space with an $s$-regular measure $μ$. We prove that if $A\subset X$ is $\varrho$-porous, then $\dim_{\mathrm{p}}(A)\le s-c\varrho^s$ where $\dim_{\mathrm{p}}$ is the packing dimension and $c$ is a positive constant which depends on $s$ and the structure constants of $μ$. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if $μ$ is a doubling measure. However, in the doubling case we find a fixed $N\subset X$ with $μ(N)=0$ such that $\dim_{\mathrm{p}}(A)\le\dim_{\mathrm{p}}(X)-c(\log\tfrac 1\varrho)^{-1}\varrho^t$ for all $\varrho$-porous sets $A\subset X\setminus N$. Here $c$ and $t$ are constants which depend on the structure constant of $μ$. Finally, we characterize uniformly porous sets in complete $s$-regular metric spaces in terms of regular sets by verifying that $A$ is uniformly porous if and only if there is $t<s$ and a $t$-regular set $F$ such that $A\subset F$.

math.CA↗