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Abstract: We study global regularity properties of Sobolev homeomorphisms on$n$-dimensional Riemannian manifolds under the assumption of $p$-integrabilityof its first weak derivatives in degree $p\geq n-1$. We prove that inversehomeomorphisms have integrable first weak derivatives. For the case $p>n$ weobtain necessary conditions for existence of Sobolev homeomorphisms betweenmanifolds. These necessary conditions based on Poincar\-e type inequality: $$\inf {c\in \mathbb R} \|u-c\mid L {\infty}M\|\leq K \|u\midL^1 {\infty}M\|. $$ As a corollary we obtain the following geometricalnecessary condition:{\em If there exists a Sobolev homeomorphisms $\phi: M \to M-$, $\phi\inW^1 pM, M-$, $p>n$, $Jx,\phi e 0$ a. e. in $M$, of compact smoothRiemannian manifold $M$ onto Riemannian manifold $M-$ then the manifold $M-$has finite geodesic diameter.}}



Author: V. Gol'dshtein, A. Ukhlov

Source: https://arxiv.org/



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