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

2015 ◽  
Vol 92 (3) ◽  
pp. 739-742 ◽  
Author(s):  
S. K. Vodop’yanov ◽  
A. O. Molchanova
Keyword(s):  
Nonlinear Elasticity ◽  
Variational Problems ◽  
Finite Distortion ◽  
Mappings With Finite Distortion

On mappings with finite length distortion on Riemann surfaces

2019 ◽  
Vol 33 ◽  
Author(s):  
Serhii Volkov ◽  
Vladimir Ryazanov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Main Tool ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Lipschitz Mappings ◽  
Geometric Function ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

scholarly journals On connection of finite distortion mappings with length distortion in $\mathbb{R}^n$

2021 ◽  
Vol 17 ◽  
pp. 112
Author(s):  
Ye.A. Sevostianov
Keyword(s):  
Finite Length ◽  
Measure Zero ◽  
Finite Distortion ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is devoted to the investigations of mappings with finite distortion in $\mathbb{R}^n$, $n \geqslant 2$. In the work it is proved that every open discrete mapping with finite distortion by Iwaniec such that the branch set of $f$ is of measure zero is a mapping with finite length distortion provided that the corresponding outer dilatation satisfies the inequality $K_O (x, f) \leqslant K(x)$ a.e., where $K(x) \in L_{loc}^{n-1}(D)$.

scholarly journals On removal of isolates singularities of open discrete mappings

2021 ◽  
Vol 15 ◽  
pp. 124
Author(s):  
Ye.A. Sevostianov
Keyword(s):  
Isolated Singularities ◽  
Finite Distortion ◽  
Mappings With Finite Distortion

We study mappings with finite distortion in $\mathbb{R}^n$, $n \geqslant 2$. For the classes of mappings with branching that satisfy modal inequalities we obtain the set of theorems on removal of isolated singularities.

scholarly journals Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians

2019 ◽  
Vol 150 (1) ◽  
pp. 41-71 ◽  
Author(s):  
Jonathan J. Bevan ◽  
Sandra Käbisch
Keyword(s):  
Nonlinear Elasticity ◽  
Variational Problems ◽  
Energy Functional ◽  
Stationary Points ◽  
Two Dimensional ◽  
Twist Maps ◽  
Regularity Results ◽  
Energy Minimizers ◽  
The Relationship ◽  
Domains With Corners

AbstractIn this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine, in particular, the relationship between the positivity of the Jacobian det ∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit explicit twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma }: \Omega \to {\open R}^2$ in a model, two-dimensional case. We exploit the Jacobian constraint $\det \nabla u_{\sigma} \gt 0$ a.e. to obtain regularity results that apply ‘up to the boundary’ of domains with corners. It is shown that the unique shear map minimizer has the properties that (i) $\det \nabla u_{\sigma }$ is strictly positive on one part of the domain Ω, (ii) $\det \nabla u_{\sigma } = 0$ necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma }$ is not continuous on the whole domain.

scholarly journals Prime ends in theory of mappings with finite distortion in the plane

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1349-1366 ◽  
Author(s):  
Denis Kovtonyuk ◽  
Igor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Boundary Behavior ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Beltrami Equations ◽  
Lipschitz Mappings ◽  
Prime Ends ◽  
The Given ◽  
Mappings With Finite Distortion ◽  
Homeomorphic Extension ◽  
Effective Conditions

In the present paper, it is studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi-Lipschitz mappings that a far-reaching extension of the known classes of isometric and quasiisometric mappings.

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