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Arkady Poliakovsky

Publications and source records attributed to Arkady Poliakovsky.

15 recordsLinked to original sources
arXiv2026

Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces

Let $X$ be a metric space and $\mu$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u \in B^r_{q,\infty}(X,\mu;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a $\sigma$-finite set with respect to the Hausdorff measure $\mathcal{H}^{s - rq}$. The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions $u$ in the fractional Sobolev space $W^{r,q}(X,\mu;Y)$, almost every point with respect to $\mathcal{H}^{s - rq}$ is an {\it average Lebesgue point} of $u$. Finally, if $Y$ is also complete, we prove that for $u \in B^r_{q,\infty}(X,\mu;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s - rq$.

math.FA↗
arXiv2025

BMO-Interpolations and Jump Detection for Functions in $BV\cap BMO$

A generalization of the John--Nirenberg inequality is established. As a consequence, local and global $BMO$--interpolation inequalities in Lorentz spaces $L^{q,\gamma}$ are obtained for the full range $0<q<\infty$ and $0<\gamma\leq\infty$. These inequalities yield interpolation results in Besov spaces, fractional Sobolev spaces, and the space $BV$, including corresponding weak spaces. As a geometric application, consequences for the jump set of functions in $BV\cap BMO$ are derived.

math.FA↗
arXiv2024

Approximations in Besov Spaces and Jump Detection of Besov Functions with Bounded Variation

In this paper, we provide a proof that functions belonging to Besov spaces $B^{r}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$, $q\in [1,\infty)$, $r\in(0,1)$, satisfy the following formula under a certain condition: \begin{equation} \label{eq:main result in abstract} \lim_{{\epsilon}\to 0^+}\frac{1}{|\ln{\epsilon}|}\left[u_{\epsilon}\right]^q_{W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{{\epsilon}\to 0^+}\int_{\mathbb{R}^N}\frac{1}{{\epsilon}^N}\int_{B_{\epsilon}(x)}\frac{|u(x)-u(y)|^q}{|x-y|^{rq}}dydx. \end{equation} Here, $\left[\cdot\right]_{W^{r,q}}$ represents the Gagliardo seminorm, and $u_{\epsilon}$ denotes the convolution of $u$ with a mollifier $\eta_{(\epsilon)}(x):=\frac{1}{\epsilon^N}\eta\left(\frac{x}{\epsilon}\right)$, $\eta\in W^{1,1}(\mathbb{R}^N),\int_{\mathbb{R}^N}\eta(z)dz=1$. Furthermore, we prove that every function $u$ in $BV(\mathbb{R}^N,\mathbb{R}^d)\cap B^{1/p}_{p,\infty}(\mathbb{R}^N,\mathbb{R}^d),p\in(1,\infty),$ satisfies \begin{multline} \lim_{\epsilon\to 0^+}\frac{1}{|\ln{\epsilon}|}\left[u_{\epsilon}\right]^q_{W^{1/q,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{{\epsilon}\to 0^+}\int_{\mathbb{R}^N}\frac{1}{{\epsilon}^N}\int_{B_{\epsilon}(x)}\frac{|u(x)-u(y)|^q}{|x-y|}dydx =\left(\int_{S^{N-1}}|z_1|~d\mathcal{H}^{N-1}(z)\right)\int_{\mathcal{J}_u} \Big|u^+(x)-u^-(x)\Big|^q d\mathcal{H}^{N-1}(x), \end{multline} for every $1<q<p$. Here $u^+,u^-$ are the one-sided approximate limits of $u$ along the jump set $\mathcal{J}_u$.

math.FA↗
arXiv2023

Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions

In this paper we analyse functions in Besov spaces $B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)$. We prove for Besov functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$ the summability of the difference between one-sided approximate limits in power $q$, $|u^+-u^-|^q$, along the jump set $\mathcal{J}_u$ of $u$ with respect to Hausdorff measure $\mathcal{H}^{N-1}$, and establish the best bound from above on the integral $\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}$ in terms of Besov constants. We show for functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$ that \begin{equation} \liminf\limits_{\varepsilon \to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for every $x$ outside of a $\mathcal{H}^{N-1}$-sigma finite set. For fractional Sobolev functions $u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ we prove that \begin{equation} \lim_{\rho\to 0^+}\fint_{B_{\rho}(x)}\fint_{B_{\rho}(x)} |u\big(z\big)-u(y)|^qdzdy=0 \end{equation} for $\mathcal{H}^{N-rq}$ a.e. $x$, where $q\in[1,\infty)$, $r\in(0,1)$ and $rq\leq N$. We prove for $u\in W^{1,q}(\mathbb{R}^N),1<q\leq N$, that \begin{equation} \lim\limits_{\varepsilon\to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for $\mathcal{H}^{N-q}$ a.e. $x\in \mathbb{R}^N$.

math.CA↗
arXiv2021

Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung

We provide answers to some questions raised in a recent work by H. Brezis, J. Van Schaftingen and Po-Lam Yung concerning the Gagliardo semi-norm $|u|_{W^{s,q}}$ computed at $s = 1$, when the strong $L^q$ is replaced by weak $L^q$. In particular, we address generalization of their results for a general domain and non-smooth functions.

math.AP↗
arXiv2018

Asymptotic behavior of the $W^{1/q,q}$-norm of mollified $BV$ functions and applications to singular perturbation problems

Motivated by results of Figalli and Jerison and Hern\'andez, we prove the following formula: \begin{equation*} \lim_{\epsilon\to 0^+}\frac{1}{|\ln{\epsilon}|}\big\|\eta_\epsilon*u\big\|^q_{W^{1/q,q}(\Omega)}= C_0\int_{J_u}\Big|u^+(x)-u^-(x)\Big|^qd\mathcal{H}^{N-1}(x), \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a regular domain, $u\in BV(\Omega)\cap L^\infty$, $q>1$ and $\eta_\epsilon(z)=\epsilon^{-N}\eta(z/\epsilon)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems.

math.AP↗
arXiv2017

Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals

Motivated by the formula, due to Bourgain, Brezis and Mironescu, \begin{equation*} \lim_{\varepsilon\to 0^+} \int_\Omega\int_\Omega \frac{|u(x)-u(y)|^q}{|x-y|^q}\,\rho_\varepsilon(x-y)\,dx\,dy=K_{q,N}\|\nabla u\|_{L^{q}}^q\,, \end{equation*} that characterizes the functions in $L^q$ that belong to $W^{1,q}$ (for $q>1$) and $BV$ (for $q=1$), respectively, we study what happens when one replaces the denominator in the expression above by $|x-y|$. It turns out that, for $q>1$ the corresponding functionals "see" only the jumps of the $BV$ function. We further identify the function space relevant to the study of these functionals, the space $BV^q$, as the Besov space $B^{1/q}_{q,\infty}$. We show, among other things, that $BV^q(\Omega)$ contains both the spaces $BV(\Omega)\cap L^\infty(\Omega)$ and $W^{1/q,q}(\Omega)$. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

math.AP↗
arXiv2015

On non-topological solutions for planar Liouville Systems of Toda-type

Motivated by the study of non abelian Chern Simons vortices of non topological type in Gauge Field Theory, we analyse the solvability of planar Liouville systems of Toda type in presence of singular sources. We identify necessary and sufficient conditions on the "flux" pair which ensure the radial solvability of the system. Since the given system includes the (integrable) 2 X 2 Toda system as a particular case, thus we recover the existence result available in this case. Our method relies on a blow-up analysis, which even in the radial setting, takes new turns compared with the single equation case.

math.AP↗
arXiv2015

Non-relativistic model of the laws of gravity and electromagnetism, invariant under the change of inertial and non-inertial coordinate systems

Under the classical non-relativistic consideration of the space-time we propose the model of the laws of gravity and Electrodynamics, invariant under the galilean transformations and moreover, under every change of non-inertial cartesian coordinate system. Being in the frames of non-relativistic model of the space-time, we adopt some general ideas of the General Theory of Relativity, like the assumption of invariance of the most general physical laws in every inertial and non-inertial coordinate system and equivalence of factious forces in non-inertial coordinate systems and the force of gravity. Moreover, in the frames of our model, we obtain that the laws of Non-relativistic Quantum Mechanics also invariant under the change of inertial or non-inertial cartesian coordinate system.

physics.gen-ph↗
arXiv2013

On the $Γ$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part II: The lower bound

In part II we constructed the lower bound, in the spirit of $Γ$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_Ω\frac{1}{\e}F\Big(\e^n\nabla^n v,...,\e\nabla v,v\Big)dx\quad\text{for}\;\; v:Ω\subset\R^N\to\R^k\;\;\text{such that}\;\; A\cdot\nabla v=0, where the function $F\geq 0$ and $A:\R^{k\times N}\to\R^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\text{curl}\, v$ and $A\cdot\nabla v=\text{div} v$). Furthermore, we studied the cases where we can easy prove the coinciding of this lower bound and the upper bound obtained in [33]. In particular we find the formula for the $Γ$-limit for the general class of anisotropic problems without a differential constraint (i.e., in the case $A:\equiv 0$).

math.AP↗
arXiv2013

Variational resolution for some general classes of nonlinear evolutions. Part I

We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the minimizer of the appropriate energy functional but also any critical point must be a solution of the corresponding evolutional system.

math.AP↗
arXiv2013

On the $Γ$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound

In Part I we construct the upper bound, in the spirit of $Γ$- $\limsup$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form $$E_\e(v):=\int_Ω\frac{1}{\e}F\Big(\e^n\nabla^n v,...,\e\nabla v,v\Big)dx\quad\text{for} v:Ω\subset\R^N\to\R^k \text{such that} A\cdot\nabla v=0,$$ where the function $F\geq 0$ and $A:\R^{k\times N}\to\R^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\text{curl}v$ and $A\cdot\nabla v=\text{div}\,v$) which includes, in particular, the problems considered in [27]. This bound is in general sharper then one obtained in [27].

math.AP↗
arXiv2011

On the $\Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part III: The energies with non local terms

We use the technique developed in [32]-[33] to construct the upper and the lower bounds for classes of problems containing non-local terms, including problems in micromagnetics and problems arising in the variational study of the Method of Vanishing Viscosity for systems of conservation laws. We reduced these problems to the problems considered in [32]-[33], with the appropriate prescribed differential constraint.

math.AP↗