Sobolev homeomorphisms and Poincare inequality

TL;DR

This paper investigates the regularity and geometric implications of Sobolev homeomorphisms between Riemannian manifolds, establishing conditions under which inverse maps are integrable and deriving necessary geometric constraints based on Poincaré inequalities.

Contribution

It proves inverse Sobolev homeomorphisms have integrable derivatives and establishes necessary conditions for their existence based on Poincaré inequalities, linking Sobolev regularity to manifold geometry.

Findings

Inverse Sobolev homeomorphisms have integrable derivatives.

Necessary conditions for Sobolev homeomorphisms depend on Poincaré inequalities.

Manifolds admitting such homeomorphisms have finite geodesic diameter.

Abstract

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.}}