Sobolev homeomorphisms and Poincare inequality

2007/12/13 by Vladimir Gol’dshtein, Alexander Ukhlov, Gol'dshtein, V. +1
Mathematics · #30C65 #46E35 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.0712.2147

openalex publication_date 2007/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥ 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 Poincaré type inequality: inf_{c∈ \mathbb R} ‖u-c| L_∞(M)‖≤ K ‖u| L¹_∞(M)‖. As a corollary we obtain the following geometrical necessary condition: \em If there exists a Sobolev homeomorphisms ϕ: M → M', ϕ∈ W¹_p(M, M'), p>n, J(x,ϕ)≠ 0 a. e. in M, of compact smooth Riemannian manifold M onto Riemannian manifold M' then the manifold M' has finite geodesic diameter.

Citations

• Convexity conditions and existence theorems in nonlinear elasticity

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