arXiv subjects

Vladimir Gol'dshtein

Publications and source records attributed to Vladimir Gol'dshtein.

At least 19 recordsLinked to original sources

arXiv2026

Notes on Electrostatics in $\mathbb R^3$

A compact formulation of electrostatics is presented. Without using constitutive relations, including the aether relations, an expression for the potential energy of a charged region under a potential function is proposed, where the charge distribution is specified by the charge potential field. Assuming that during a virtual motion of the charge, the virtual work expended by the field is equal to minus the time derivative of the potential energy, an expression for the mechanical force functional is derived. The force functional contains the action of an asymmetric active stress field D_{i}E_{j}, the skew-symmetric part of which is the mechanical couple density.

math-ph↗

arXiv2026

Sobolev $(p,q)$-extension operators and Neumann eigenvalues

In this article, we consider $(p,q)$-extension operators, $1 < q \le p < \infty$, on Sobolev spaces. Based on composition operators on Sobolev spaces, we construct the extension operators in outward cuspidal domains with estimates of their norms. Using these $(p,q)$-extension operators, we prove estimates for the non-linear Neumann eigenvalues of the $p$-Laplace operator in outward cuspidal domains.

math.AP↗

arXiv2026

On Force Interactions for Electrodynamics-Like Theories

A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples.

math-ph↗

arXiv2025

Some Poincaré--Sobolev inequalities for differential forms

We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology and Sobolev type inequalities. The~main results are estimates for the norms of the embedding operators for $q=p$ and $p>\frac{n-1}{k-1}$ in the~Euclidean $r$-ball $B(r)$ and its bi-Lipschitz images. We also study the~compactness of such operators.

math.DG↗

arXiv2025

On the weighted Steklov eigenvalue problems in outward cuspidal domains

In this article, we investigate the weighted Steklov eigenvalue problem and the weighted Schrödinger--Steklov eigenvalue problem in outward cuspidal domains. We prove the solvability of these spectral problems in both linear and non-linear cases.

math.AP↗

arXiv2025

Multipole Distributions and Hyper-Flux Fields

We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property $Π$ from the point of view of continuum mechanics. Classically, $Π$ is the electric charge, but the theory is not limited to electrostatics. The proposed framework allows a simple computation of the bound "charges" and bound multipoles of lower orders. In addition, if the property $Π$ has a potential function in the sense described below, a general expression for the mechanical force (power) functional acting on bodies containing the property is presented. Finally, using a similar viewpoint, we consider hyper-fluxes -- flux fields of tensorial order greater than one -- and show that moving multipoles (in particular, a moving dielectric) give rise to hyper-fluxes.

math-ph↗

arXiv2024

Geometric theory of composition operators on Sobolev spaces

In this paper, we present the basic concepts of the geometric theory of composition operators on Sobolev spaces. The main objects of the theory are topological mappings which generate bounded embedding operators on Sobolev spaces by the composition rule. This theory is in some sense a "generalization" of the theory of quasiconformal mappings, but the theory of composition operators is oriented to its applications to the Sobolev embedding theorems, the spectral theory of elliptic operators and continuum mechanics problems.

math.AP↗

arXiv2024

Notes on Optimal Flux Fields

For a given region, and specified boundary flux and density rate of an extensive property, the optimal flux field that satisfies the balance conditions is considered. The optimization criteria are the $L^{p}$-norm and a Sobolev-like norm of the flux field. Finally, the capacity of the region to accommodate various boundary fluxes and density rates is defined and analyzed.

math.AP↗

arXiv2023

On differentiability of Sobolev functions with respect to the Sobolev norm

We study connections between the $W^1_p$-differentiability and the $L_p$-differentiability of Sobolev functions. We prove that, $W^1_p$-differentiability implies the $L_p$-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the $W^1_p$-differentiability of Sobolev functions $\cp_p$-almost everywhere.

math.AP↗

arXiv2023

On the theory of generalized quasiconformal mappings

We study generalized quasiconformal mappings in the context of the inverse Poletsky inequality. We consider the local behavior and the boundary behavior of mappings with the inverse Poletsky inequality. In particular, we obtain logarithmic Hölder continuity for such classes of mappings.

math.CV↗

arXiv2023

On the principal frequency of non-homogeneous membranes

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $Ω\subset \mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincaré inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.

math.AP↗

arXiv2022

On the boundary behavior of weak $(p,q)$-quasiconformal mappings

Let $Ω$ and $\widetildeΩ$ be domains in the Euclidean space $\mathbb R^n$. We study the boundary behavior of weak $(p,q)$-quasiconformal mappings, $φ:Ω\to \widetildeΩ$, $n-1<q\leq p<n$. The suggested method is based on the capacitary distortion properties of the weak $(p,q)$-quasiconformal mappings.

math.AP↗

arXiv2022

A Lipschitz version of de Rham theorem for $L_p$-cohomology

We focus our attention on the de Rham operators' underlying properties which are specified by intrinsic effects of differential geometry structures. And then we apply the procedure of regularization in the context of Lipschitz version of de Rham calculus on metric simplicial complexes with bounded geometry.

math.DG↗

arXiv2022

Composition operators on Sobolev spaces and Ball's classes

In this paper we study geometric aspects of Ball's classes in the context of nonlinear elasticity problems. The suggested approach is based on the characterization of Ball's classes $A_{q,q'}(Ω)$ in terms of composition operators on Sobolev spaces. This characterization allows us to obtain the volume compression (distortion) estimates of topological mappings of Ball's classes. We prove also that Sobolev homeomorphisms of Ball's classes which possess the Luzin $N$-property are absolutely continuous with respect to capacity.

math.AP↗

arXiv2021

Principal frequencies of free non-homogeneous membranes and Sobolev extension operators

Using the quasiconformal mappings theory and Sobolev extension operators, we obtain estimates of principal frequencies of free non-homogeneous membranes. The suggested approach is based on connections between divergence form elliptic operators and quasiconformal mappings. Free non-homogeneous membranes we consider as circular membranes in the quasiconformal geometry associated with non-homogeneity of membranes. As a consequence we get a connection between principal frequencies of free membranes and the smallest-circle problem (initially suggested by J.~J. Sylvester in 1857).

math.AP↗

arXiv2021

Composition operators on Sobolev spaces and weighted moduli inequalities

In this paper we study connections between composition operators on Sobolev spaces and mappings defined by $p$-moduli inequalities ($p$-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on corresponding Sobolev spaces and inverse, composition operators on Sobolev spaces imply weighted moduli inequalities.

math.AP↗

arXiv2021

Quasiconformal Whitney Partition

Whitney partition is a very important concept in modern analysis. We discuss here a quasiconformal version of the Whitney partition that can be usefull for Sobolev spaces.

math.FA↗

arXiv2020

Estimates of Dirichlet eigenvalues of divergent elliptic operators in non-Lipschitz domains

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $Ω\subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincaré-Sobolev inequalities.

math.AP↗