Global Invertibility for Orientation-Preserving Sobolev Maps via Invertibility on or Near the Boundary
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Global Invertibility for Orientation-Preserving Sobolev Maps via Invertibility on or Near the Boundary Stefan Krömer Communicated by I. Fonseca
Abstract By a result of Ball (Proc R Soc Edinb Sect A Math 88:315–328, 1981. https:// doi.org/10.1017/S030821050002014X), 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.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1. Basic Notation and Terminology . . . . . . . . . . 2. Constraints Related to Global Invertibility . . . . . . . 2.1. Approximate Invertibility . . . . . . . . . . . . . . 2.2. The Ciarlet–Neˇcas Condition and Condition (INV) 2.3. Maps of Topological Degree at Most One . . . . . 3. The Degree: Basic Notation and Properties . . . . . . . 4. The Degree and Approximate Invertibility . . . . . . . 5. The Degree and Orientation Preserving Maps . . . . . . 5.1. Strictly Orientation Preserving Maps . . . . . . . .
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ˇ and the AusThis research was supported by the Czech Science Foundation (GA CR) trian Science Fund (FWF) through the bilateral Grant 19-29646L (Large Strain Challenges in Materials Science), and through the associated MSMT-WTZ bilateral travel Grant 8J19AT013
S. Krömer 5.2. Strictly Orientation Preserving Maps of Degree One . . . . . . . . 6. Global Invertibility in W 1, p . . . . . . . . . . . . . . . . . . . . . . . 6.1. Ball’s Global Invertibility Revisited . . . . . . . . . . . . . . . . . 6.2. Improved Invertibility Exploiting Finite Distortion . . . . . . . . . 6.3. Existence of Homeomorphic Minimizers . . . . . . . . . . . . . . 6.4. Homeomorphic Extension for C 1 Functions on Lipschitz Domains References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction A classical problem in Nonlinear Elasticity is to determine whether a Sobolev map y : Ω → Rd on a bounded domain Ω ⊂ Rd , is invertible in a suitable sense. In this context, the map y describes the deformation of an elastic solid occupying Ω in its undeformed state. In this model, lack of invertibilty cor
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