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Alexander Menovschikov

Publications and source records attributed to Alexander Menovschikov.

13 recordsLinked to original sources
arXiv2026

Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants

We study a nonlinear Neumann-to-Steklov limit generated by a family of interior weights concentrating at the boundary. On a class of admissible possibly irregular domains obtained from the unit ball by trace-compatible Sobolev homeomorphisms, we consider the first nontrivial weighted \((p,q)\)-Neumann eigenvalue with respect to a concentrating bulk weight \(γ_a\). We prove that, as \(a\to0\), these eigenvalues converge to the corresponding weighted \((p,q)\)-Steklov eigenvalue with boundary weight \(β\). Moreover, normalized minimizers converge, up to subsequences, strongly in \(W^{1,p}\) to Steklov minimizers. Equivalently, the best constants in the weighted Poincaré inequalities converge to the best constants in the weighted trace inequalities; in fact, a quantitative convergence estimate is obtained in the subcritical trace range.

math.AP↗
arXiv2026

Compactness of weighted Sobolev trace operators and non-linear Steklov problems

We prove the compactness of weighted Sobolev trace operators in outward cuspidal domains by using composition operators on Sobolev spaces. This result allows us to formulate the non-linear Steklov problem in outward cuspidal domains in a correct functional setting and to establish the existence of its non-trivial solution.

math.AP↗
arXiv2025

Nonlinear Neumann eigenvalues in outward cuspidal domains with weighted measure

We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These embeddings give the solvability of the Neumann spectral problem in this setting and provide estimates for the corresponding weighted Neumann eigenvalues.

math.AP↗
arXiv2024

Zero-extension convergence and Sobolev spaces on changing domains

We extend the definition of weak and strong convergence to sequences of Sobolev-functions whose underlying domains themselves are converging. In contrast to previous works, we do so without ever assuming any sort of reference configuration. We then develop the respective theory and counterparts to classical compactness theorems from the fixed domain case. Finally, we illustrate the usefulness of these definitions with some examples from applications and compare them to other approaches.

math.AP↗
arXiv2023

Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings

We provide the estimates for the constant in the weighted Poincaré inequality for a special class of planar domains and weights. Based on this, we prove the lower bounds for the first non-zero eigenvalue $μ_ρ$ of the Neumann Laplacian with density $ρ$. These estimates depend on the density function and the geometry of the domain. In particular, it is shown, that $μ_ρ$ can be made arbitrarily large by changing the mass density of the domain.

math.AP↗
arXiv2022

Composition operators on Sobolev spaces, $Q$-mappings and weighted Sobolev inequalities

In this paper we give connections between mappings which generate bounded composition operators on Sobolev spaces and $Q$-mappings. On this base we obtain measure distortion properties $Q$-homeomorphisms. Using the composition operators on Sobolev spaces we obtain weighted Sobolev inequalities with special weights which are Jacobians of $Q$-mappings.

math.AP↗
arXiv2022

Decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces

The notion of decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces is introduced. We show that these operators generalize the composition operator, in sense that a mapping is replaced by a binary relation. The necessary and sufficient conditions for the boundedness of those operators are the main results of the paper.

math.FA↗
arXiv2021

An extended variational theory for nonlinear evolution equations via modular spaces

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. Firstly, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality paring can be introduced and a generalised computational chain rule holds. Secondly, we prove well-posedness in an extended variational sense for evolution equations, without relying on any reflexivity assumption and any polynomial requirement on the nonlinearity. Finally, we discuss several important applications that can be addressed in this framework: these cover, but are not limited to, equations in Musielak-Orlicz-Sobolev spaces, such as variable exponent, Orlicz, weighted Lebesgue, and double-phase spaces.

math.AP↗
arXiv2020

Sobolev space of functions valued in a monotone Banach family

We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics.

math.FA↗
arXiv2017

Bounded operators on mixed norm Lebesgue spaces

We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given. For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy-Steklov type operators.

math.FA↗