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Anastasia Molchanova

Publications and source records attributed to Anastasia Molchanova.

13 recordsLinked to original sources
arXiv2026

Derivation of variational membrane models in the context of anisotropic nonlocal hyperelasticity

Motivated by the analysis of thin structures, we study the variational dimension reduction of hyperelastic energies involving nonlocal gradients to an effective membrane model. When rescaling the thin domain, isotropic interaction ranges naturally become anisotropic, leading to the development of a theory for anisotropic nonlocal gradients with direction-dependent interaction ranges. Unlike existing nonlocal derivatives with finite horizon, which are defined via interaction kernels supported on balls of positive radius, our formulation is based on ellipsoidal interaction regions whose principal radii may vanish independently. This yields a unified framework that interpolates between fully nonlocal, partially nonlocal, and purely local models. Employing these tools, we present a $Γ$-convergence analysis for the nonlocal thin-film energies. The limit functional retains the structural form of the classical membrane energy, and the classical local model is recovered precisely when all interaction radii vanish.

math.AP↗
arXiv2024

Linearization in magnetoelasticity

Starting from a model of nonlinear magnetoelasticity where magnetization is defined in the Eulerian configuration while elastic deformation is in the Lagrangean one, we rigorously derive a linearized model that coincides with the standard one that already appeared in the literature, see, e.g., DeSimone \& James (2002) and where the zero-stress strain is quadratic in the magnetization. The relation of the nonlinear and linear model is stated in terms of the $Γ$-convergence and convergence of minimizers.

math.AP↗
arXiv2023

A new example for the Lavrentiev phenomenon in Nonlinear Elasticity

We present a new example for the Lavrentiev phenomenon in context of nonlinear elasticity, caused by an interplay of the elastic energy's resistance to infinite compression and the Ciarlet-Nečas condition, a constraint preventing global interpenetration of matter on sets of full measure.

math.AP↗
arXiv2023

Weak limit of homeomorphisms in $W^{1,n-1}$: invertibility and lower semicontinuity of energy

Let $Ω$, $Ω'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colonΩ\toΩ'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the Lusin (N) condition and $\sup_m \int_Ω(|Df_m|^{n-1}+A(|\text{cof} Df_m|)+ϕ(J_f))<\infty$, where $A$ and $φ$ are positive convex functions. Let $f$ be a weak limit of $f_m$ in $W^{1,n-1}$. Provided certain growth behaviour of $A$ and $φ$, we show that $f$ satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

math.FA↗
arXiv2023

Differentiability almost everywhere of weak limits of bi-Sobolev homeomorphisms

This paper investigates the differentiability of weak limits of bi-Sobolev homeomorphisms. Given $p>n-1$, consider a sequence of homeomorphisms $f_k$ with positive Jacobians $J_{f_k} >0$ almost everywhere and $\sup_k(\|f_{k}\|_{W^{1,n-1}} + \|f_{k}^{-1}\|_{W^{1,p}}) <\infty$. We prove that if $f$ and $h$ are weak limits of $f_k$ and $f_k^{-1}$, respectively, with positive Jacobians $J_f>0$ and $J_h>0$ a.e., then $h(f(x))=x$ and $f(h(y))=y$ both hold a.e.\ and $f$ and $h$ are differentiable almost everywhere.

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↗
arXiv2021

Equilibria of charged hyperelastic solids

We investigate equilibria of charged deformable materials via the minimization of an electroelastic energy. This features the coupling of elastic response and electrostatics by means of a capacitary term, which is naturally defined in Eulerian coordinates. The ensuing electroelastic energy is then of mixed Lagrangian-Eulerian type. We prove that minimizers exist by investigating the continuity properties of the capacitary terms under convergence of the deformations

math.AP↗
arXiv2021

Regularity of the inverse mapping in Banach function spaces

We study the regularity properties of the inverse of a bilipschitz mapping $f$ belonging $W^m X_{\text{loc}}$, where $X$ is an arbitrary Banach function space. Namely, we prove that the inverse mapping $f^{-1}$ is also in $W^m X_{\text{loc}}$. Furthermore, the paper shows that the class of bilipschitz mappings in $W^m X_{\text{loc}}$ is closed with respect to composition and multiplication.

math.FA↗
arXiv2020

On grand Sobolev spaces and pointwise description of Banach function spaces

We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy-Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets (of finite or infinite measure), and provide pointwise description of them.

math.FA↗
arXiv2019

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

Let $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(Ω,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For $p\leq n-1$ we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain.

math.CA↗
arXiv2019

A note on the continuity of minors in grand Lebesgue spaces

We present a simple proof of the continuity, in the sense distributions, of the minors of the differential matrices of mappings belonging to grand Sobolev spaces. Such function spaces were introduced in connection with a problem on minimal integrability of the Jacobian and are useful in certain aspects of geometric function theory and partial differential equations.

math.FA↗
arXiv2018

Interpolation between H\" older and Lebesgue spaces with applications

Classical interpolation inequality of the type $\|u\|_{X}\leq C\|u\|_{Y}^θ\|u\|_{Z}^{1-θ}$ is well known in the case when $X$, $Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing norms $\|\cdot\|_{Y}$ or $\|\cdot\|_{X}$ by suitable H\" older semi-norm. We shall even prove sharper version involving weak Lorentz norm. We apply this result to prove the Gagliardo--Nirenberg inequality for a wider scale of parameters.

math.FA↗