arXiv subjects

S. K. Vodop'yanov

Publications and source records attributed to S. K. Vodop'yanov.

3 recordsLinked to original sources

arXiv2018

Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity

We show that a sufficient condition for the weak limit of a sequence of $W^1_q$-homeomorphisms with finite distortion to be almost everywhere injective for $q \geq n-1$, can be stated by means of composition operators. Applying this result, we study nonlinear elasticity problems with respect to these new classes of mappings. Furthermore, we impose loose growth conditions on the stored-energy function for the class of $W^1_n$-homeomorphisms with finite distortion and integrable inner as well as outer distortion coefficients.

math.FA↗

arXiv2017

Sobolev $W_{p}^{1}(\mathbb{R}^{n})$ spaces on $d$-thick closed subsets of $\mathbb{R}^{n}$

Let $S \subset \mathbb{R}^{n}$ be a~closed set such that for some $d \in [0,n]$ and $\varepsilon > 0$ the~$d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S \cap Q(x,r)) \geq \varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x \in S$ with side length $2r \in (0,2]$. For every $p \in (1,\infty)$, denote by $W_{p}^{1}(\mathbb{R}^{n})$ the classical Sobolev space on $\mathbb{R}^{n}$. We give an~intrinsic characterization of the restriction $W_{p}^{1}(\mathbb{R}^{n})|_{S}$ of the space $W_{p}^{1}(\mathbb{R}^{n})$ to~the set $S$ provided that $p > \max\{1,n-d\}$. Furthermore, we prove the existence of a bounded linear operator $\operatorname{Ext}:W_{p}^{1}(\mathbb{R}^{n})|_{S} \to W_{p}^{1}(\mathbb{R}^{n})$ such that $\operatorname{Ext}$ is right inverse for the usual trace operator. In particular, for $p > n-1$ we characterize the trace space of the Sobolev space $W_{p}^{1}(\mathbb{R}^{n})$ to the closure $\overlineΩ$ of an arbitrary open path-connected set~$Ω$. Our results extend those available for $p \in (1,n]$ with much more stringent restrictions on~$S$.

math.FA↗

arXiv2015

Variational problems of nonlinear elasticity theory in certain classes of mappings with finite distortion

We study the problem of minimizing the functional $$ I(φ)=\int\limits_Ω W(x,Dφ)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $φ\in W^1_n(Ω)$ and growth conditions on the integrand $W(x,F)$. To compensate for that, we impose the finite distortion condition and the condition $\frac{|Dφ(x)|^n}{J(x,φ)} \leq M(x) \in L_{s}(Ω)$, $s>n-1$, on the characteristic of distortion. On assuming that the integrand $W(x,F)$ is polyconvex and coercive, we obtain an~existence theorem for the problem of minimizing the functional $I(φ)$ on a new family of admissible deformations. KEYWORDS: functional minimization problem, nonlinear elasticity, mapping with finite distortion, polyconvexity.

math.AP↗