Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary

Abstract

By a result of John Ball (1981), a locally orientation preserving Sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension. As shown here for any dimension, the conclusions of Ball's theorem and related results can be reached while completely avoiding the problem of homeomorphic extension. For suitable domains, it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps. An application in Nonlinear Elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed. As a tool in the proofs, strictly orientation-preserving maps and their global invertibility properties are studied from a purely topological point of view.

44 pages. Slightly extended version with clarifications and corrections

References in corpus (3)

• Monotone Sobolev Mappings of planar domains and surfaces
• A phase-field approach to Eulerian interfacial energies
• A Schoenflies extension theorem for a class of locally bi-Lipschitz homeomorphisms

Cited by in corpus (5)

• Weak limit of homeomorphisms in $W^{1,n-1}$ and (INV) condition
• Linearized von Kármán theory for incompressible magnetoelastic plates
• Sobolev regularity of the inverse for minimizers of the neo-Hookean energy satisfying condition INV
• Approximation of planar Sobolev $W^{2,1}$ homeomorphisms by Piecewise Quadratic Homeomorphisms and Diffeomorphisms
• Invertibility of Sobolev maps through approximate invertibility at the boundary and tangential polyconvexity