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Register a new userApproximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms
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Given a Sobolev homeomorphism $fin W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.
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Let $Omegasubseteqmathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $fin W^{1}X(Omega,mathcal{R}^2)$ be a homeomorphism between $Omega$ and $f(Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(Omega,mathcal{R}^2)$.
Given any $f$ a locally finitely piecewise affine homeomorphism of $Omega subset rn$ onto $Delta subset rn$ in $W^{1,p}$, $1leq p < infty$ and any $epsilon >0$ we construct a smooth injective map $tilde{f}$ such that $|f-tilde{f}|_{W^{1,p}(Omega,rn)} < epsilon$.
We show that given a homeomorphism $f:GrightarrowOmega$ where $G$ is a open subset of $mathbb{R}^2$ and $Omega$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincare inequality, it holds $fin BV_{operatorname{loc}}(G,Omega)$ if and only $f^{-1}in BV_{operatorname{loc}}(Omega,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $fin W^{1,1}_{operatorname{loc}}(G,Omega)$ if and only if $f^{-1}in W^{1,1}_{operatorname{loc}}(Omega,G)$.
The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $bullet$ the generating function of such a foliation is C1 ; $bullet$ the foliation is H{o}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnold-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugated to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the 2-dimensional annulus has Arnold-Liouville coordinates and then provide examples of strange Lipschitz foliations in smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.This article is a part of another preprint of the authors, entitled On the transversal dependence of weak K.A.M. solutions for symplectic twist maps, after rewriting ant adding of the H{o}lder part.
We present a classification of strict limits of planar BV homeomorphisms. The authors and S. Hencl showed in a previous work cite{CHKR} that such mappings allow for cavitations and fractures singularities but fulfill a suitable generalization of the INV condition. As pointed out by J. Ball cite{B}, these features are physically expected by limit configurations of elastic deformations. In the present work we develop a suitable generalization of the emph{no-crossing} condition introduced by De Philippis and Pratelli in cite{PP} to describe weak limits of planar Sobolev homeomorphisms that we call emph{BV no-crossing} condition, and we show that a planar mapping satisfies this property if and only if it can be approximated strictly by homeomorphisms of bounded variations.
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