Sobolev homeomorphisms and Poincare inequality

Details Sobolev homeomorphisms and Poincare inequality Sobolev homeomorphisms and Poincare inequality

2007-12-13

arxiv.org/abs/0712.2147v2

Mathematics

Functional Analysis

In the first version, there was an inaccuracy in Theorem 4. In the revised version added additional assumptions

Scientific paper

We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincar\'e type inequality: $$ \inf_{c\in \mathbb R} \|u-c\mid L_{\infty}(M)\|\leq K \|u\mid L^1_{\infty}(M)\|. $$ As a corollary we obtain the following geometrical necessary condition: {\em If there exists a Sobolev homeomorphisms $\phi: M \to M'$, $\phi\in W^1_p(M, M')$, $p>n$, $J(x,\phi)\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}

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