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Global Invertibility for Orientation-Preserving Sobolev Maps via Invertibility on or Near the Boundary

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.

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References

  1. Alexander, J.W.: An example of a simply connected surface bounding a region which is not simply connected. Proc. Natl. Acad. Sci. 10(1), 8–10, 1924. https://doi.org/10.1073/pnas.10.1.8

    ADS  Article  MATH  Google Scholar 

  2. Alexander, J.W.: On the subdivision of 3-space by a polyhedron. Proc. Natl. Acad. Sci. USA 10(1), 6–8, 1924

    ADS  Article  Google Scholar 

  3. Ball, J.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. Sect. A Math. 88, 315–328, 1981. https://doi.org/10.1017/S030821050002014X.

    MathSciNet  Article  MATH  Google Scholar 

  4. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(4), 337–403, 1976/77. https://doi.org/10.1007/BF00279992

  5. Barchiesi, M., Henao, D., Mora-Corral, C.: Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity. Arch. Ration. Mech. Anal. 224(2), 743–816, 2017. https://doi.org/10.1007/s00205-017-1088-1

    MathSciNet  Article  MATH  Google Scholar 

  6. Bouchala, O., Hencl, S., Molchanova, A.: Injectivity almost everywhere for weak limits of Sobolev homeomorphisms. Journal of Functional Analysis 279(7), 108658, 2020. https://doi.org/10.1016/j.jfa.2020.108658

    MathSciNet  Article  MATH  Google Scholar 

  7. Brown, M.: Locally flat imbeddings of topological manifolds. Ann. Math. 2(75), 331–341, 1962. https://doi.org/10.2307/1970177

    MathSciNet  Article  MATH  Google Scholar 

  8. Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 173–188, 1987. https://doi.org/10.1007/BF00250807

    MathSciNet  Article  MATH  Google Scholar 

  9. Daneri, S., Pratelli, A.: A planar bi-Lipschitz extension theorem. Adv. Calc. Var. 8(3), 221–266, 2015. https://doi.org/10.1515/acv-2012-0013

    MathSciNet  Article  MATH  Google Scholar 

  10. Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications, vol. 2. Oxford Lecture Series in Mathematics and Its ApplicationsOxford University Press, New York 1995

    MATH  Google Scholar 

  11. Fox, R.H., Artin, E.: Some wild cells and spheres in three-dimensional space. Ann. Math. 2(49), 979–990, 1948. https://doi.org/10.2307/1969408

    MathSciNet  Article  MATH  Google Scholar 

  12. Giacomini, A., Ponsiglione, M.: Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinburgh Sect. A 138(5), 1019–1041, 2008. https://doi.org/10.1017/S0308210507000121

    MathSciNet  Article  MATH  Google Scholar 

  13. Gong, J.: A Schoenflies extension theorem for a class of locally bi-Lipschitz homeomorphisms. Michigan Math. J. 60(3), 507–523, 2011. https://doi.org/10.1307/mmj/1320763046

    MathSciNet  Article  MATH  Google Scholar 

  14. Grandi, D., Kružík, M., Mainini, E., Stefanelli, U.: A phase-field approach to Eulerian interfacial energies. Arch. Ration. Mech. Anal. 234(1), 351–373, 2019. https://doi.org/10.1007/s00205-019-01391-8

    MathSciNet  Article  MATH  Google Scholar 

  15. Henao, D., Mora-Corral, C.: Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity. Arch. Ration. Mech. Anal. 197(2), 619–655, 2010. https://doi.org/10.1007/s00205-009-0271-4

    MathSciNet  Article  MATH  Google Scholar 

  16. Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion, vol. 2096. Lecture Notes in Mathematics. Springer, Cham 2014

  17. Hencl, S., Rajala, K.: Optimal assumptions for discreteness. Arch. Ration. Mech. Anal. 207(3), 775–783, 2013. https://doi.org/10.1007/s00205-012-0574-8

    MathSciNet  Article  MATH  Google Scholar 

  18. Iwaniec, T., Onninen, J.: Monotone Sobolev mappings of planar domains and surfaces. Arch. Ration. Mech. Anal. 219(1), 159–181, 2016. https://doi.org/10.1007/s00205-015-0894-6

    MathSciNet  Article  MATH  Google Scholar 

  19. Iwaniec, T., Onninen, J.: Radó-Kneser–Choquet theorem for simply connected domains (\(p\)-harmonic setting). Trans. Am. Math. Soc. 371(4), 2307–2341, 2019. https://doi.org/10.1090/tran/7348

    Article  MATH  Google Scholar 

  20. Kałamajska, A., Krömer, S., Kružík, M.: Sequential weak continuity of null Lagrangians at the boundary. Calc. Var. Part. Differ. Equ. 49(3–4), 1263–1278, 2014. https://doi.org/10.1007/s00526-013-0621-9

    MathSciNet  Article  MATH  Google Scholar 

  21. Kielhöfer, H.: Bifurcation theory, Applied Mathematical Sciences, vol. 156, 2nd edn. Springer, New York 2012. https://doi.org/10.1007/978-1-4614-0502-3. (An introduction with applications to partial differential equations)

  22. Krömer, S., Valdman, J.: Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms on the boundary (in preparation).

  23. Krömer, S., Valdman, J.: Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms. Math. Mech. Solids 24(11), 3644–3673, 2019. https://doi.org/10.1177/1081286519851554

    MathSciNet  Article  Google Scholar 

  24. Luukkainen, J., Väisälä, J.: Elements of Lipschitz topology. Ann. Acad. Sci. Fenn. Ser. A I Math. 3(1), 85–122, 1977. https://doi.org/10.5186/aasfm.1977.0315

    MathSciNet  Article  MATH  Google Scholar 

  25. Marcus, M., Mizel, V.J.: Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems. Bull. Am. Math. Soc. 79, 790–795, 1973. https://doi.org/10.1090/S0002-9904-1973-13319-1

    MathSciNet  Article  MATH  Google Scholar 

  26. Martio, O., Ziemer, W.P.: Lusin’s condition (N) and mappings with nonnegative Jacobians. Michigan Math. J. 39(3), 495–508, 1992. https://doi.org/10.1307/mmj/1029004603

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. Mazur, B.C.: On embeddings of spheres. Acta Math. 105, 1–17, 1961. https://doi.org/10.1007/BF02559532

    MathSciNet  Article  MATH  Google Scholar 

  28. Meisters, G.H., Olech, C.: Locally one-to-one mappings and a classical theorem on schlicht functions. Duke Math. J. 30, 63–80, 1963

    MathSciNet  Article  Google Scholar 

  29. Milnor, J.: On manifolds homeomorphic to the \(7\)-sphere. Ann. Math. 2(64), 399–405, 1956. https://doi.org/10.2307/1969983

    MathSciNet  Article  MATH  Google Scholar 

  30. Molchanova, A., Vodopyanov, S.: Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity. Calc. Var. Part. Differ. Equ. 59(1), 17, 2019. https://doi.org/10.1007/s00526-019-1671-4

    MathSciNet  Article  MATH  Google Scholar 

  31. Müller, S.: Higher integrability of determinants and weak convergence in \(L^1\). J. Reine Angew. Math. 412, 20–34, 1990. https://doi.org/10.1515/crll.1990.412.20

    MathSciNet  Article  MATH  Google Scholar 

  32. Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131(1), 1–66, 1995. https://doi.org/10.1007/BF00386070

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. Müller, S., Spector, S.J., Tang, Q.: Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal. 27(4), 959–976, 1996. https://doi.org/10.1137/S0036141094263767

    MathSciNet  Article  MATH  Google Scholar 

  34. Outerelo, E., Ruiz, J.M.: Mapping Degree Theory, vol. 108. Graduate Studies in MathematicsAmerican Mathematical Society, Providence 2009. https://doi.org/10.1090/gsm/108

    Book  MATH  Google Scholar 

  35. Rajala, K.: Remarks on the Iwaniec–Šverák conjecture. Indiana Univ. Math. J. 59(6), 2027–2039, 2010. https://doi.org/10.1512/iumj.2010.59.3946

    MathSciNet  Article  MATH  Google Scholar 

  36. Rajala, K.: Reshetnyak’s theorem and the inner distortion. Pure Appl. Math. Q. 7(2, Special Issue: In honor of Frederick W. Gehring, Part 2), 411–424, 2011. https://doi.org/10.4310/PAMQ.2011.v7.n2.a8

  37. Reshetnyak, Y.G.: Space mappings with bounded distortion (Translations of Mathematical Monographs), vol. 73. American Mathematical Society, Providence 1989. (Translated from the Russian by H. H. McFaden)

  38. Šverák, V.: Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100(2), 105–127, 1988. https://doi.org/10.1007/BF00282200

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. Tang, Q.: Almost-everywhere injectivity in nonlinear elasticity. Proc. R. Soc. Edinburgh Sect. A 109(1–2), 79–95, 1988. https://doi.org/10.1017/S030821050002669X

    MathSciNet  Article  MATH  Google Scholar 

  40. Titus, C.J., Young, G.S.: The extension of interiority, with some applications. Trans. Am. Math. Soc. 103, 329–340, 1962. https://doi.org/10.2307/1993663

    MathSciNet  Article  MATH  Google Scholar 

  41. Tukia, P.: The planar Schönflies theorem for Lipschitz maps. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 5(1), 49–72, 1980. https://doi.org/10.5186/aasfm.1980.0529

    MathSciNet  Article  MATH  Google Scholar 

  42. Villamor, E., Manfredi, J.J.: An extension of Reshetnyak’s theorem. Indiana Univ. Math. J. 47(3), 1131–1145, 1998. https://doi.org/10.1512/iumj.1998.47.1323

    MathSciNet  Article  MATH  Google Scholar 

  43. Vodopyanov, S., Goldshtein, V.: Quasiconformal mappings and spaces of functions with generalized first derivatives. Siber. Math. J. 17, 399–411, 1976. https://doi.org/10.1007/BF00967859

    Article  Google Scholar 

  44. Weinstein, A.: A global invertibility theorem for manifolds with boundary. Proc. R. Soc. Edinburgh Sect. A 99(3–4), 283–284, 1985. https://doi.org/10.1017/S030821050001430X

    MathSciNet  Article  MATH  Google Scholar 

  45. Whyburn, G.T.: Topological Analysis, vol. 23. Princeton Mathematical SeriesPrinceton University Press, Princeton 1958

    MATH  Google Scholar 

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Correspondence to Stefan Krömer.

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Communicated by I. Fonseca

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This research was supported by the Czech Science Foundation (GA ČR) and the Austrian 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

Appendices

A The Problem of Homeomorphic Extension

When working with injective continuous maps, it is good to keep in mind the following two well-known facts.

Lemma A.1

Let XZ be topoogical spaces and \(y:X\rightarrow Z\) continuous and injective, and suppose that X is compact. Then \(y:X\rightarrow y(X)\) is a homeomorphism, where y(X) is endowed with the trace topology of Z.

Proof

By continuity of y, y(X) is also compact. Open sets in X and y(X), respectively, are thus exactly the complements of compact sets. Since \(y:X\rightarrow y(X)\) is bijective and maps compact sets to compact sets, it therefore also maps open sets in X to open sets in y(X). \(\quad \square \)

The statement above does not mean that y maps opens sets in X to open sets in Z, because y(X) is usually not open in Z. In \({\mathbb {R}}^d\), more can be said:

Theorem A.2

Let \(\varOmega \subset {\mathbb {R}}^d\) be open, \(\varOmega \ne \emptyset \), and let \(y:\varOmega \rightarrow {\mathbb {R}}^n\) be injective and continuous. Then \(n\ge d\). Moreover, \(y(\varOmega )\) is open in \({\mathbb {R}}^n\) if and only if \(n=d\). For \(n=d\), \(y:\varOmega \rightarrow y(\varOmega )\) is a homeomorphism.

Proof

This is a combination of the openness (invariance of domain) and invariance of dimension theorems based on the topological degree, see [10, Thm. 3.30, Cor. 3.31 and Cor. 3.32] (for example). The last assertion is a consequence of the others which also hold for arbitrary open subsets of \(\varOmega \), thereby proving that y maps open subsets of \(\varOmega \) to open sets in \({\mathbb {R}}^n={\mathbb {R}}^d\). \(\quad \square \)

A.1 Homeomorphic Extension Versus Schoenflies Extension

We are here mainly interested in homeomorphic extension for functions given on the boundary of some domain in \({\mathbb {R}}^d\). In the most general form, this problem reads as follows:

Problem A.3

(Homeomorphic extension problem in \({\mathbb {R}}^d\)) Let \(\varOmega \subset {\mathbb {R}}^d\) a bounded domain, and suppose that \(y: \partial \varOmega \rightarrow {\mathbb {R}}^d\) is continuous and injective. Is there a homeomorphism \(h:{\bar{\varOmega }} \rightarrow h({\bar{\varOmega }})\subset {\mathbb {R}}^d\) such that \(h=y\) on \(\partial \varOmega \)?

Additional assumptions on the topological nature of \(\varOmega \) and \(\partial \varOmega \) are typically added, as it is well-known that simple counterexamples exist when \(\varOmega \) is topologically complicated. For instance, on an annulus, homeomorphic extension is impossible if the winding numbers of y on the two boundary pieces are not the same (for example, one clockwise and the other counterclockwise).

A close relative is the following question which is much more widely studied in the literature:

Problem A.4

(Schoenflies extension problem) Let \(y: S^{d-1}\rightarrow S^d\) continuous and injective. Is there a homeomorphism \(h:S^d\rightarrow S^d\) such that \(h(S^{d-1})=\varSigma ^{d-1}{:}{=}y(S^{d-1})\), where \(S^{d-1}\subset S^d\) is interpreted as the equator of the d-dimensional sphere \(S^d\subset {\mathbb {R}}^{d+1}\)?

The Schoenflies extension problem imposes a restriction on the topological type of admissible domains (one of the half-spheres separated by the equator)—it must be a topological ball—which is also commonly used for the homeomorphic extension problem. Apart from that, the two problems are essentially equivalent.

Proposition A.5

(Schoenflies versus homeomorphic extension)

Suppose that \(\varOmega \subset {\mathbb {R}}^d\) is a bounded domain such that there exists a homeomorphism \(\gamma :{\bar{\varOmega }}\rightarrow \bar{H}\), where \(H\subset S^{d}\subset {\mathbb {R}}^{d+1}\) is one of the two hemispheres of \(S^d\) separated by the “equator” \(S^{d-1}\). Moreover, let \(y: S^{d-1}\rightarrow S^d\) continuous and injective and let \(\delta :S^d\rightarrow {\mathbb {R}}^d\cup \{\infty \}\) be a homeomorphism with \(\delta (y(S^{d-1}))\subset {\mathbb {R}}^d\), where \({\mathbb {R}}^d\cup \{\infty \}\) denotes the one-point compactification of \({\mathbb {R}}^d\). Then we have the following for \(\tilde{y}{:}{=}\delta \circ y \circ \gamma \):

  1. (i)

    If there exists Schoenflies extension \(h:S^d\rightarrow S^d\) of y as in Problem A.4, then a homeomorphic extension \(\tilde{h}\) of \(\tilde{y}\) as in Problem A.3 exists, too.

  2. (ii)

    Conversely, if no Schoenflies extension \(h:S^d\rightarrow S^d\) of y exists, a homeomorphic extension also fails to exist for one of the following two maps:

    1. (a)

      \(\tilde{y}:\partial \varOmega \rightarrow {\mathbb {R}}^d\), or

    2. (b)

      \(\hat{y}{:}{=}\tilde{y}\circ \iota :\partial \hat{\varOmega }\rightarrow {\mathbb {R}}^d\), with \(\hat{\varOmega }\) denoting the bounded connected component of \({\mathbb {R}}^d{\setminus } \iota (\partial \varOmega )\).

Here, \(\iota : {\mathbb {R}}^d\cup \{\infty \}\rightarrow {\mathbb {R}}^d\cup \{\infty \}\), \(\iota (x){:}{=}\left| x-x_0\right| ^{-2}(x-x_0)\), is the inversion map with respect to a point \(x_0\in {\mathbb {R}}^d\); for (ii), we choose an arbitrary but fixed \(x_0\in \varOmega \).

Proof

(i) \({\tilde{\varrho }}{:}{=}h^{-1}\circ y:S^{d-1}\rightarrow S^{d-1}\) is a homeomorphism of the equator \(S^{d-1}\) onto itself. It has an explicit “radial” homeomorphic extension \(\varrho :S^d\rightarrow S^d\). Using cylindrical coordinates \((x,t)\in S^{d-1}\times [-1,1]\), it is given by

$$\begin{aligned} \varrho \big ((1-t^2)^\frac{1}{2}x,t\big ){:}{=}\big ((1-t^2)^{\frac{1}{2}} {\tilde{\varrho }}(x),t\big )\in S^d\subset {\mathbb {R}}^d\times {\mathbb {R}}. \end{aligned}$$

Using the inversion map \(\iota \) with respect to a point \(x_0\) in the bounded connected component of \(\tilde{y}(\partial \varOmega )\subset {\mathbb {R}}^d\), we now define \(\tilde{h}{:}{=}\delta \circ h\circ \varrho \circ \gamma \) or \(\tilde{h}{:}{=}\iota \circ \delta \circ h \circ \varrho \circ \gamma \). One of the two options satisfies \(\infty \notin \tilde{h}({\bar{\varOmega }})\), and for this choice, \(\tilde{h}|_{{\bar{\varOmega }}}\) is a homeomorphic extension of \(\tilde{y}\) in the sense of Problem A.3.

(ii) If a Schoenflies extension \(h:S^d\rightarrow S^d\) of y does not exists, then extension already fails in one of the two hemispheres of \(S^d\) separated by \(S^{d-1}\), either \(\bar{H}=\gamma ({\bar{\varOmega }})\) or \(S^d{\setminus } H\). (Otherwise, the extensions to the hemispheres can be glued to a Schoenflies extension, after first matching their parametrization of \(y(S^{d-1})\) using the radial homeomorphic extension of the proof of (i).) Accordingly, for either \(\tilde{y}\) or \(\hat{y}=\tilde{y}\circ \iota \), where \(\iota \) is taken with respect to an \(x_0\in \varOmega \), there exists no homeomorphism defined on the closure of \(\varOmega \) or \({\hat{\varOmega }}\), respectively, who maps the boundary of its domain to \(\tilde{y}(\partial \varOmega )=\hat{y}(\partial {\hat{\varOmega }})\). In particular, either \(\tilde{y}\) or \(\hat{y}\) has no homeomorphic extension to its domain in the sense of Problem A.3. \(\quad \square \)

A. 2 Known Results and Counterexamples

For \(d=2\), the answer to Problem A.4 is affirmative, given by the classical Schoenflies Theorem. Extension theorems for more regular classes of invertible functions are also known in this case, for instance bi-Lipschitz extensions [9, 41]. Recently, an extension result (also) valid in the class of Sobolev homeomorphisms has been established in [19, Theorem 4 and Corollary 5]. This is based on p-harmonic extension and even smooth in \(\varOmega \).

For \(d\ge 3\), the situation is significantly more complicated. In general, a Schoenflies extension can fail to exist, for instance in case of Alexander’s horned sphere [1]. However, the result can be recovered for \(d\ge 3\) if the embedded sphere \(\varSigma ^{d-1}{:}{=}y(S^{d-1})\) is locally flat:

Theorem A.6

(Generalized Schoenflies Theorem [7, Theorem 4])

Let \(\varSigma ^{d-1}\subset S^d\) be a homeomorphic embedding of \(S^{d-1}\) which is locally flat in the following sense:

$$\begin{aligned} \begin{aligned}&\mathrm{For}\, \mathrm{each }\, x_0\in \varSigma ^{d-1}, \mathrm{there}\, \mathrm{exists}\, \mathrm{a}\, \mathrm{neighborhood }\, V \, \mathrm{of }\, x_0\, \mathrm{in}\, S^d \, \mathrm{and}\\&\mathrm{a} \, \mathrm{homeomorphism }\, \zeta :V\rightarrow \zeta (V)\subset S^d \, \mathrm{s.t.}\, \zeta (V\cap \varSigma ^{d-1})=\zeta (V)\cap S^{d-1}, \end{aligned} \end{aligned}$$

where \(S^{d-1}\) is interpreted as the equator of \(S^{d}\). Then \(\varSigma ^{d-1}\) is flat, that is, there is a homeomorphism \(h:S^d\rightarrow S^d\) such that \(h(\varSigma ^{d-1})=S^{d-1}\).

There are also variants of the Generalized Schoenflies Theorem that require higher regularity of \(\varSigma ^{d-1}\) instead of assuming a locally flat embedding. In particular, this is possible for the piecewise affine (polyhedral) [2] or diffeomorphic [27] case. As pointed out in [24, Example 3.10 (5)] for \(d=3\), bi-Lipschitz regularity is not enough.

Remark A.7

(Homeomorphic extension may fail for \(d\ge 3\)) In view of Proposition A.5, [24, Example 3.10 (5)] also entails that for \(d=3\), homeomorphic extension is in general impossible even if the given boundary homeomorphism \(y:\partial \varOmega \rightarrow \varSigma ^{d-1}{:}{=}y(\partial \varOmega )\) is bi-Lipschitz.

Remark A.8

The example of [24] is based on the Fox-Artin arc [11, Example 1.1], a bi-Lipschitz embedding of a compact interval into \({\mathbb {R}}^3\) whose image has a complement which is not simply connected. By thickening it, surrounding the original interval by a domain consisting of two thin cones back-to-back with tips at the two end points of the interval, the self-similar construction yields a bi-Lipschitz mapping of the domain boundary onto a surface in \({\mathbb {R}}^3\). This surface is a topological 2-sphere, and from the Fox-Artin arc, it inherits that the unbounded component of its complement is not simply connected. In particular, a Schoenflies extension (after identifying \({\mathbb {R}}^3\cup \{\infty \}\) with \(S^3\)) is impossible because its existence would imply that both halves of \(S^3\) separated by the surface are topological 3-balls which are simply connected.

On the other hand, if we look for a solution of Problem A.4 when the embedding of the sphere is known to be locally flat with higher regularity given in the whole neighborhood of its image \(\varSigma ^{d-1}\), then this regularity sometimes can be carried over to a suitable extension. In particular, this is possible in the bi-Lipschitz case [24, Theorem 7.7], or for the second order bi-Sobolev homeomorphisms where both the function and its inverse are in \(W^{2,p}\) with \(1\le p<d\) [13]. As far as I know, there is no comparable result for bi-Sobolev homeomorphisms in \(W^{1,p}\) (yet?), only the theory for maps with finite distortion [16] which is conceptually closer to regularity theory than to extension results. In the diffeomorphic category, extensions starting from locally flat embeddings face another obstacle in higher dimensions, the possible existence of exotic spheres, for example for \(d=8\) (7-dimensional spheres) [29, Theorem 3.4].

Homeomorphic Extension for \(C^1\) Functions on Lipschitz Domains

As shown in Proposition A.5, a solution to Problem A.4 can be used to build homeomorphic extensions of maps \(y|_{\partial \varOmega }\), at least if \(\varOmega \) is homeomorphic to the closed unit ball. A more practical application in the same spirit is given below, using Theorem A.6 to obtain a homeomorphic extension of a \(C^1\)-deformation on a Lipschitz domain which is invertible on the boundary. Despite the similarity, it does not directly follow from the Schoenflies extension for a \(C^1\) map outlined in [27, p.11], because we would first have to transform the given Lipschitz domain to the unit ball. This is possible, but the transformation is only bi-Lipschitz and we would lose the crucial \(C^1\) regularity (cf. Remark A.7).

Theorem A.9

Suppose that \(\varOmega \subset {\mathbb {R}}^d\) is a Lipschitz domain such that \({\bar{\varOmega }}\) is homeomorphic to the closed unit ball. Moreover, let \(y\in C^1({\bar{\varOmega }};{\mathbb {R}}^d)\) such that \(y|_{\partial \varOmega }\) is injective and \(\det \nabla y\ne 0\) on \(\partial \varOmega \). Then \(y|_{\partial \varOmega }\) admits a homeomorphic extension to \({\bar{\varOmega }}\).

Proof

The proof is based on Theorem A.6 and Proposition A.5 (i). To apply the theorem, we identify \(S^d\) with the one-point compactification \({\mathbb {R}}^d\cup \{\infty \}\) of \({\mathbb {R}}^d\). In this sense, \({\mathbb {R}}^d\subset S^d\) (homeomorphically embedded), and y maps \({\bar{\varOmega }}\) to \({\mathbb {R}}^d\subset S^d\), and \(\varSigma ^{d-1}{:}{=}y(\partial \varOmega )\) is a homeomorphic embedding of a topological \((d-1)\)-dimensional sphere into \({\mathbb {R}}^d\subset S^d\). To see that this embedding is also locally flat in the sense of Theorem A.6, it suffices to define local bi-Lipschitz extensions of \(y\in C^1({\bar{\varOmega }};{\mathbb {R}}^d)\) in a neighborhood of each boundary point \(x_0\in \partial \varOmega \). Here, notice that \(\partial \varOmega \) is locally the graph of a Lipschitz function. This implies that \(\partial \varOmega \) is locally flat, and we may therefore assume that the homeomorphism mapping \(\varOmega \) to the unit ball is defined on a whole neighborhood of \({\bar{\varOmega }}\).

Since \(\partial \varOmega \) is Lipschitz, we can choose a cylidrical neigborhood of the form \(C_\varepsilon (x_0){:}{=}D_\varepsilon (x_0)+(-\varepsilon ,\varepsilon )\nu \subset {\mathbb {R}}^d\) with a unit vector \(\nu =\nu (x_0)\in {\mathbb {R}}^d\) and a \((d-1)\)-dimensional disc \(D_\varepsilon (x_0)\) of radius \(\varepsilon \), centered at \(x_0\) and perpendicular to \(\nu \). For \(\varepsilon >0\) small enough and an appropriate choice of \(\nu \), \(\partial \varOmega \cap C_\varepsilon (x_0)\) can be represented as the graph of a Lipschitz function \(g:D_\varepsilon (x_0)\rightarrow (-\varepsilon ,\varepsilon )\), such that \(\varOmega \cap C_\varepsilon (x_0)=\left\{ x'+t\nu \,\left| \,t<g(x')\right. \right\} \). We can now extend \(y|_{\partial \varOmega }\) to a function \(\hat{y}:C_\varepsilon (x_0)\rightarrow {\mathbb {R}}^d\) by setting

$$\begin{aligned} \hat{y}(x'+t\nu ){:}{=}y(x'+g(x')\nu )+(t-g(x'))Dy(x_0)\nu . \end{aligned}$$

for \(x'\in D_\varepsilon (x_0)\) and \(t\in (-\varepsilon ,\varepsilon )\). Close to \(x_0\), this extension divides \(C_\varepsilon (x_0)\) into surfaces of the form \(\partial \varOmega +s\nu \) (parametrized by \(x'+g(x')\in \partial \varOmega \) and \(s=t-g(x')\)) and maps each such surface onto \(y(\partial \varOmega )+s Dy(x_0)\nu \), a shifted copy of \(y(\partial \varOmega )\).

To see that \(\hat{y}\) is bi-Lipschitz in a neighborhood of \(x_0\), the key observation is the following: Just like \(\nu \) and \(\partial \varOmega \), \(Dy(x_0)\nu \) and the surface \(y(\partial \varOmega )\) always form an angle bounded away from zero as long as we remain close enough to \(x_0\), because Dy is continuous and \(Dy(x_0)\) is invertible. As an immediate consequence, \(\partial \varOmega \rightarrow {\mathbb {R}}^d\), \(\sigma \mapsto y(\sigma )+s Dy(x_0)\) is injective near \(x_0\) for each s, and in a small enough neighborhood V of \(y(x_0)\) in \({\mathbb {R}}^d\), we also have that

$$\begin{aligned} V\cap [y(\partial \varOmega )+s Dy(x_0)\nu ]\cap [y(\partial \varOmega )+s_2 Dy(x_0)\nu ]=\emptyset \quad \text {for}~s_1\ne s_2. \end{aligned}$$

Hence, \(\hat{y}\) is injective near \(x_0\). Further details are omitted.

Theorem A.6 now gives us a Schoenfliess extension of \(\varSigma ^{d-1}{:}{=}y(\partial \varOmega )\subset {\mathbb {R}}^d\cup \{\infty \}\cong S^d\), and by Proposition A.5 (i), this implies the existence of a homeomorphic extension of \(y|_{\partial \varOmega }\) to \({\bar{\varOmega }}\). \(\quad \square \)

B Counterexamples for Domains with Holes

The following examples illustrate that the assumption that \({\mathbb {R}}^d{\setminus } \partial \varOmega \) has only two connected components cannot be dropped in Theorem 4.2. For simplicity, they are all constructed for \(d=2\), but they have straightforward equivalents in higher dimensions, still using domains with holes. In particular, it does not really matter whether \(\varOmega \) is simply connected or not. In both examples, the explicit values asserted for the degree are always taken at a suitable regular value of y with just one pre-image \(x_0\in \varOmega \), and are therefore given as the sign of \(\det \nabla y(x_0)\). Geometric intuition provides a good heuristic, observing whether or not the local deformation is orientation-preserving. If yes, the sign is positive, otherwise negative.

Example B.1

Take the annulus \(\varOmega {:}{=}B_2(0){\setminus } {\bar{B}}_1(0)\subset {\mathbb {R}}^2\) and for \(x\in \varOmega \) consider \(y\in W^{1,\infty }(\varOmega ;{\mathbb {R}}^2)\),

$$\begin{aligned} y(x){:}{=}2\frac{\left| x\right| -1}{\left| x\right| }x+\frac{2-\left| x\right| }{\left| x\right| }\left( x+(3,0)\right) . \end{aligned}$$

As defined, y keeps the outer boundary \(\partial B_2(0)\) fixed while translating the inner boundary onto \(y_1(\partial B_1(0))=(3,0)+\partial B_1(0)\). In particular, \(y|_{\partial \varOmega }\) is invertible, but it maps \(\partial \varOmega \) to two circles that lie outside of each other. Now, \(\mathcal {B}=B_2(0)\cup [(3,0)+B_1(0)]\) and \(\sigma \) changes sign; more precisely, \({\text {deg}}(y;\varOmega ;(0,0))=1\) while \({\text {deg}}(y;\varOmega ;(3,0))=-1\).

It is also not enough to have that \(\partial \varOmega \) is connected:

Example B.2

Take a fixed unit vector \(e\subset {\mathbb {R}}^2\) and truncated open cones of the form

$$\begin{aligned} \hat{V}(\alpha ,r){:}{=}O(r,\alpha )\cup \left\{ x\in {\mathbb {R}}^2\,\left| \,x\cdot e> (1-\alpha )\left| x\right| ,~\left| x\right| <r\right. \right\} ,~~ \alpha ,r>0, \end{aligned}$$

where \(O(r,\alpha )\subset {\mathbb {R}}^s\) denotes the unique open ball which touches the surface of the unbounded cone tangentially at \(\left| x\right| =r\). Consequently, \(V_{\alpha ,r}\) has a boundary of class \(C^1\) everywhere except at its tip in the origin. We create a domain by removing a smaller cone from a bigger one sharing the same tip:

$$\begin{aligned} \varOmega {:}{=}V_2{\setminus } {\bar{V}}_1,\quad \text {where }V_s{:}{=}\hat{V}\big (\tfrac{s}{3},s\big ). \end{aligned}$$

As a first step, we now consider a map \(y\in W^{1,\infty }(\varOmega ;{\mathbb {R}}^2)\) which keeps the outer part of the boundary fixed while flipping the inner part outside, with affine interpolation on suitable rays in between. For its explicit definition, the flip is realized by the reflection R across \(\{x\cdot e=0\}\), \(R x{:}{=}x-2(x\cdot e)e\), and we use the (nonlinear) projections Q(x) and P(x) onto the inner and the outer boundary, respectively, along lines perpendicular to the inner boundary \(\partial V_{1}\): For all \(x\in \varOmega {\setminus } \{0\}\),

$$\begin{aligned} Q(x)\in \partial V_{1},~~P(x)\in \partial V_{2}~~\text {and}~~ P(x)-Q(x)\perp \partial V_{1}~~\text {at}~Q(x). \end{aligned}$$

Notice that \(Q,P:\varOmega \rightarrow {\mathbb {R}}^2\) are Lipschitz and thus in \(W^{1,\infty }\) (even \(C^1\) away from \(x=0\)), and both converge to the origin as \(\left| x\right| \rightarrow 0\), \(x\in \varOmega \). As illustrated in Fig. 3, we now can define

$$\begin{aligned} \begin{aligned} y(x){:}{=} \frac{\left| x-Q(x)\right| }{\left| P(x)-Q(x)\right| } P(x) + \frac{\left| P(x)-x\right| }{\left| P(x)-Q(x)\right| } R[Q(x)]&\\ = \left( 1+\frac{\left| P(x)-x\right| }{\left| P(x)-Q(x)\right| }\right) \big (R[Q(x)]-P(x)\big )&. \end{aligned} \end{aligned}$$

The latter representation shows that y is Lipschitz also at the origin.

Fig. 3
figure 3

\(\varOmega \) and the definition of y in Example B.2

This construction does not yet contradict the assertion of Theorem 4.2, because as a matter of fact, \({\text {deg}}(y;\varOmega ;\cdot )=1\) on both components of \(\mathcal {B}({\mathbb {R}}^d{\setminus } y(\partial \varOmega )) =V_2\cup (-V_1)\). In any case, with a second deformation that squeezes the line orthogonal to e in the image to the origin and simultaneously reflects the half-space \(\{x\cdot e<0\}\) containing \(-V_1\) across the line in direction e, we can make the degree change sign at the value \(-e\in -V_1\) while keeping it fixed at \(e\in V_2\). More precisely, for

$$\begin{aligned} {\hat{y}}{:}{=}F\circ y\quad \text {with}~F(z){:}{=}(z\cdot e) e+(z\cdot e)(z\cdot e^\perp )e^\perp , \end{aligned}$$

we have that \(e,-e\in \mathcal {B}({\mathbb {R}}^d{\setminus } \hat{y}(\partial \varOmega ))\). Moreover, F keeps the line in direction e including those two points fixed, and they are regular values for both F and y. Hence, \({\text {deg}}(\hat{y};\varOmega ;e)=+{\text {deg}}(y;\varOmega ;e)=1\) and \({\text {deg}}(\hat{y};\varOmega ;-e)=-{\text {deg}}(y;\varOmega ;-e)=-1\), because \(\det \nabla F(-e)<0<\det \nabla F(e)\).

Remark B.3

Starting with a domain with several holes, a similar construction as in Example B.1 with a subsequent orientation-preserving deformation can also cause the images of holes to be stacked inside of each other. In fact, this way, with \(\left| n\right| \) holes for any given \(n\in {\mathbb {Z}}\), we can get a deformation y invertible on \(\partial \varOmega \), such that \({\text {deg}}(y;\varOmega ;\cdot )\) attains the value n somewhere. This also works in context of Example B.2 if we have several small conical holes that all meet at the tip of the big outer cone.

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Krömer, S. Global Invertibility for Orientation-Preserving Sobolev Maps via Invertibility on or Near the Boundary. Arch Rational Mech Anal 238, 1113–1155 (2020). https://doi.org/10.1007/s00205-020-01559-7

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