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 orientationpreserving maps and their global invertibility properties are studied from a purely topological point of view.
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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 1929646L (Large Strain Challenges in Materials Science), and through the associated MSMTWTZ 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 wellknown facts.
Lemma A.1
Let X, Z 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 wellknown 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 \(y: S^{d1}\rightarrow S^d\) continuous and injective. Is there a homeomorphism \(h:S^d\rightarrow S^d\) such that \(h(S^{d1})=\varSigma ^{d1}{:}{=}y(S^{d1})\), where \(S^{d1}\subset S^d\) is interpreted as the equator of the ddimensional 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 halfspheres 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” \(S^{d1}\). Moreover, let \(y: S^{d1}\rightarrow S^d\) continuous and injective and let \(\delta :S^d\rightarrow {\mathbb {R}}^d\cup \{\infty \}\) be a homeomorphism with \(\delta (y(S^{d1}))\subset {\mathbb {R}}^d\), where \({\mathbb {R}}^d\cup \{\infty \}\) denotes the onepoint compactification of \({\mathbb {R}}^d\). Then we have the following for \(\tilde{y}{:}{=}\delta \circ y \circ \gamma \):

(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.

(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:

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

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

(a)
Here, \(\iota : {\mathbb {R}}^d\cup \{\infty \}\rightarrow {\mathbb {R}}^d\cup \{\infty \}\), \(\iota (x){:}{=}\left xx_0\right ^{2}(xx_0)\), 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) \({\tilde{\varrho }}{:}{=}h^{1}\circ y:S^{d1}\rightarrow S^{d1}\) is a homeomorphism of the equator \(S^{d1}\) onto itself. It has an explicit “radial” homeomorphic extension \(\varrho :S^d\rightarrow S^d\). Using cylindrical coordinates \((x,t)\in S^{d1}\times [1,1]\), it is given by
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 \(S^{d1}\), 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 \(y(S^{d1})\) 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 biLipschitz 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 pharmonic 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 \(\varSigma ^{d1}{:}{=}y(S^{d1})\) is locally flat:
Theorem A.6
(Generalized Schoenflies Theorem [7, Theorem 4])
Let \(\varSigma ^{d1}\subset S^d\) be a homeomorphic embedding of \(S^{d1}\) which is locally flat in the following sense:
where \(S^{d1}\) is interpreted as the equator of \(S^{d}\). Then \(\varSigma ^{d1}\) is flat, that is, there is a homeomorphism \(h:S^d\rightarrow S^d\) such that \(h(\varSigma ^{d1})=S^{d1}\).
There are also variants of the Generalized Schoenflies Theorem that require higher regularity of \(\varSigma ^{d1}\) 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 \(d=3\), biLipschitz 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 \(y:\partial \varOmega \rightarrow \varSigma ^{d1}{:}{=}y(\partial \varOmega )\) is biLipschitz.
Remark A.8
The example of [24] is based on the FoxArtin arc [11, Example 1.1], a biLipschitz 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 backtoback with tips at the two end points of the interval, the selfsimilar construction yields a biLipschitz mapping of the domain boundary onto a surface in \({\mathbb {R}}^3\). This surface is a topological 2sphere, and from the FoxArtin 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 3balls 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 \(\varSigma ^{d1}\), then this regularity sometimes can be carried over to a suitable extension. In particular, this is possible in the biLipschitz case [24, Theorem 7.7], or for the second order biSobolev 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 biSobolev 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 \(d=8\) (7dimensional 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 biLipschitz 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 onepoint 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 \(\varSigma ^{d1}{:}{=}y(\partial \varOmega )\) is a homeomorphic embedding of a topological \((d1)\)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 biLipschitz 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 \((d1)\)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
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 \(s=tg(x')\)) 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 biLipschitz 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
Hence, \(\hat{y}\) is injective near \(x_0\). Further details are omitted.
Theorem A.6 now gives us a Schoenfliess extension of \(\varSigma ^{d1}{:}{=}y(\partial \varOmega )\subset {\mathbb {R}}^d\cup \{\infty \}\cong S^d\), 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 preimage \(x_0\in \varOmega \), 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 orientationpreserving. 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 \(y\in W^{1,\infty }(\varOmega ;{\mathbb {R}}^2)\),
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
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:
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 \(\{x\cdot e=0\}\), \(R x{:}{=}x2(x\cdot e)e\), 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 \(x\in \varOmega {\setminus } \{0\}\),
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
The latter representation shows that y is Lipschitz also at the origin.
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 halfspace \(\{x\cdot e<0\}\) 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
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 orientationpreserving 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 OrientationPreserving Sobolev Maps via Invertibility on or Near the Boundary. Arch Rational Mech Anal 238, 1113–1155 (2020). https://doi.org/10.1007/s00205020015597
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DOI: https://doi.org/10.1007/s00205020015597