INVERSE FUNCTION THEOREMS AND JACOBIANS OVER METRIC SPACES

We present inversion results for Lipschitz maps f : Ω ⊂ R → (Y, d) and stability of inversion for uniformly convergent sequences. These results are based on the Area Formula and on the l.s.c. of metric Jacobians.


Introduction
The question of inverting maps is recurrent and important in many circumstances, both theoretical or applied ones. Also the literature on the subject is very spread (linear and non-linear functional analysis, differential e/o integral equations, mathematical economics, etc.) and reflecting different approaches and points of view.
Our motivation on invertibility issues relies on mathematical variational models of continuum mechanics. In the setting of nonlinear elasticity, in the undeformed state the material body occupies a bounded open set Ω ⊂ R N . Then one usually looks for minimizers u : Ω → R N of the stored energy I(u) = Ω W (∇u) dx in an admissible class K of deformations u. In this framework, the invertibility of deformations u ∈ K corresponds to the physical assumption of impenetrability of matter. The variational approach to this kind of problems leads naturally to treat stability of invertibility with respect to suitable notions of convergence. The question is then to find conditions under which the limit map of a sequence of invertible maps is itself invertible. This basic and fundamental question has of course many possible answers. In this paper we are interested in the case of maps f : Ω ⊂ R N → f (Ω) where the target f (Ω) lies possibly in a metric space (Y, d). Our main motivation in considering this question comes from [19] where the notion of transport plan as in Optimal Mass Transportation Theory is proposed as a weak notion of material deformation. In such case (see [19] for the details) it results useful to consider an energy functional on maps f : Ω ⊂ R N → (Y, d). Precisely, in [19] Y is the set of probability measures endowed with the Wasserstein metric. In this model, invertibility of maps over a metric space was fundamental to obtain lower semicontinuity of the energy functional. It is then natural to consider the applicability of inverse function theorem for maps with metric space targets, as opposed to the usual setting of Euclidean spaces. This necessitates a shift from degree theory methods (which are standard in the Euclidean case) toward coarser, and therefore more robust, estimates based on volume growth.
For maps over R N a widely used tool is the Topological Degree Theory. However, for maps over metric spaces no such theory is available. Observe that also the notion of fixed point has no meaning in this setting. This is a main obstruction in dealing with maps over metric spaces. On the other hand, many concepts of Geometric Measure Theory are well established in this framework ( [2,3,23,24,28]). Following the observation of [20] in which some inversion results are obtained by using the area formula and lower semicontinuity of Jacobians, here we investigate inversion results based on Geometric Measure Theory for Lipschitz maps f : Ω → (Y, d).
1.1. Description of the results. In Section 2 we provide an overview of some approaches to invertibility based on the Area Formula and on l.s.c. of Jacobians. Many results in the literature deal with global invertibility obtained by local invertibility. In the framework of this paper, we observe that in fact considering open maps is enough. The point is that the so called Invariance Domain Theorem, which state that a continuous locally invertible map is an open map, is peculiar of the Euclidean setting and anyway it does not hold in general. Therefore, the notion of open map seems more appropriate in this metric setting. Outside of the Euclidean setting, i.e. without the benefit of the Invariance Domain Theorem, this seems the natural way to proceed.
A recurrent notion in literature, especially in the Sobolev setting, is that of open and discrete maps (see [30,33,32,25,26]). We also find that this kind of maps is useful from a metric point of view. Actually, this notion is crucial (see Lemma 3.14 and Theorem 3.15) to provide easy proofs of l.s.c. of metric Jacobians. Hence we provide a results of stability for invertible maps also valid over a metric target. Our main result (see Theorem 3.16) establishes that the uniform limit f of a sequence of equi-Lipschitz invertible maps is also invertible provided f is open and discrete.
The l.s.c. results for the metric Jacobians are based on l.s.c. of the Hausdorff measure with respect to Hausdorff convergence of sets. A general l.s.c. result for the metric Jacobians would probably require more subtle studies of l.s.c. with respect to Hausdorff convergence. In Section 4 we discuss invertibility of maps related with the so called quasiisometries introduced by John and also considered in [19].

Local toward global invertibility
is the multilplicity function. This is the case of course for f Lipschitz. It is possible to use the Area Formula (1) in different ways in order to obtain global invertibility of f . Let x 1 = x 2 be such that f (x 1 ) = f (x 2 ). Consider two small disjoint balls B 1 := B(x 1 , r 1 ), B 2 := B(x 2 , r 2 ). By the Invariance Domain Theorem we get that f is an open map. Therefore, the set V : is an open non-empty set. Moreover, it results N (y, f, Ω) ≥ 2 for every y ∈ V . By using (1) we obtain (2) 2H To get a contradiction we have to estimate the above Jacobian integral in terms of H N (V ). Actually, there are different possible approaches.
Therefore, if M < N √ 2 we get invertibility of f . (ii) Ciarlet-Necas: In [9] it is considered the condition which of course holds true with equality sign for invertible maps. Under (3), by (2) we estimate leading to invertibility of f . By using the area formula (see [2,23,24,28]) we get Therefore, the opposite inequality is always true also in a metric space setting, whenever the area formula holds true. (iii) Stability: In [20] it is contained a discussion about stability of invertibility. In particular it is observed that invertibility of a limit map f of a sequence f n of invertible maps can be obtained by using the lower semicontinuity of Jacobian's integrals under uniform convergence (see [1]). Indeed, fixed ε > 0, by uniform convergence we find f n (A) ⊂ f (A) ε for large n, where B ε denotes the ε-neighborhood of the set B. By (2) we obtain Letting ε → 0 we get (see also Lemma 3.11) where we used that ∂V ⊂ ∂f (B 1 ) ∪ ∂f (B 2 ) and that f is open and Lipschitz, so that H n (∂V ) = 0. (iv) Quasi-isometries: Let f be an (m − M )-quasi-isometry, namely such that The metric derivatives D + , D − are related to inversion properties, see [22,16,7] for an account on these maps. The constants m, M , roughly speaking, provides estimates of pointwise Lipschitz behavior of f from below and above respectively. By using (2), and given the volume control condition Therefore, if M m < N √ 2 we get invertibility of f . Observe that in the approaches stated above, local invertibility is not actually needed. The fact that f is an open map is enough to conclude, since the main point relies in finding the open set V such that H N (V ) > 0.

Invertibility over metric spaces
We address the question of invertibility for Lipschitz maps where Lip([0, 1], Y ) denotes the set of Lipschitz maps of [0, 1] over Y , while |γ|(t) denotes the metric derivative defined by h which exists for a.e. t due to the Lipschitz condition (see [5] for the basic properties). Any curve γ achieving the minimum value in (4) is called geodesic. Therefore, the metric space Y is geodesic if any distance d(x, y) can be obtained as the length of a shortest path (geodesic) connecting x and y.
Moreover, we shall assume that the balls of Y are somehow continuous, namely that every continuous path connecting interior and exterior points of a ball B intersects the boundary ∂B. In terms of parametrisation of a Lipschitz curve γ : ∈B, then there exists 0 < s < t such that γ(s) ∈ ∂B.

3.1.
Open and discrete maps. The Invariance Domain Theorem is peculiar of R N and heavily relies on Topological Degree Theory. However, such result does not hold in general metric spaces. Therefore, to our purpose it seems better to handle directly with open maps.
In [27] , such balls are sometimes called normal neighborhoods.
We also say that f is normal if ∂f (B) ⊂ f (∂B) for every ball B. We have the following , for an open set U ⊂ X . Consider a normal neighborhood B 0 := B(x 0 , r). We claim that y 0 is an interior point of f (B 0 ). If not, then y 0 ∈ ∂f (B 0 ) ⊂ f (∂B 0 ). Therefore y 0 = f (x) with x ∈ ∂B 0 , contradicting the discreteness of f .
Therefore, for large n, we have y n = f (x n ) with x n / ∈ B. Considering a subsequence if necessary, passing to the limit we find y 0 = f (x) with x / ∈ B contradicting the injectivity of f .
If not, then y 0 ∈ ∂f (B 0 ). Therefore, we find a sequence y n → y 0 with y n / ∈ f (B 0 ). Since f (Ω) is open, as in the proof of the above Lemma 3.2, we get . By uniform convergence we have z n → y 0 . On the other hand, y n := f n (x 0 ) ∈ f n (B 0 ) also converges to y 0 . Claim: we may assume that z n ∈ ∂f n (B 0 ).
Indeed, if not consider a point y n realizing If such points y n are not interior points, by the continuity assumption on the balls of Y , we would find points of f n (B 0 ) with lower distance from z n . Therefore we have we also have y n → y 0 .
Since y n = f n (x n ) with x n ∈ ∂B 0 , passing to a subsequence if necessary, we find y 0 = f (x) with x ∈ ∂B 0 , contradicting the discreteness of f .
Proof. Suppose by contradiction that f −1 (y) contains an accumulation point x 0 . Therefore, we get x n → x 0 , with x n ∈ f −1 (y) and The volume estimate considered in Section 2 are devoted to get N (y, f, Ω) = 1 a.e. which is then compared with the condition of open map for f . We get the following Proof. Let x 1 = x 2 be two points in Ω such that f (x 1 ) = f (x 2 ). Consider two small disjoint balls B 1 := B(x 1 , r 1 ), an open non-empty set. Moreover, it results N (y, f, Ω) ≥ 2 for every y ∈ V , which is a contradiction since H N is non-trivial.
Suppose by contradiction to have x 1 , x 2 ∈ Ω such that f (x 1 ) = f (x 2 ) = y 0 . We may assume x 1 ∈ Ω, x 2 ∈ ∂Ω. Let B 1 be a small ball around x 1 . Since f is Lipschitz and open we find a small radius r > 0 such that f (B(x 2 , r)  For the basic properties of covering maps we refer for instance to [14]. We mention the following Lemma 3.8. Let X be a geodesic metric space satisfying the continuity condition on balls. Let X be locally compact and locally pathwise connected. Let Y be locally pathwise connected and locally simply connected. Let f n : X → Y be local homeomorphisms and f : X → Y be an open and discrete map. If f n → f (locally) uniformly, then for every x 0 ∈ X there exists a neighborhood x 0 ∈ U such that every f n is an homeomorphism over U .
Proof. Let x 0 ∈ X be fixed. By uniform convergence, we may assume that f n → f in B(x 0 , s) with B(x 0 , s) ⊂ X compact. Since f is discrete we may assume that f (x) = f (x 0 ) for every x ∈ B(x 0 , s). As in [25,Proposition 7], we look for a common neighborhood of x 0 in which the functions f n are all simultaneously invertible. Setting Claim: It is possible to choose r sufficiently small so that U 1 is a normal neighborhood of x 0 and f (U 1 ) = B(y 0 , r).
It remains to check that U 1 is normal and f (U 1 ) = B(y 0 , r).
By uniform convergence, we may assume that d Y (f n (x), f (x)) < r 3 for every x ∈ B(x 0 , s). Since d Y (y n , y 0 ) < r 3 , there exists a connected component Q n of x 0 contained in Letting j → +∞ we get Since U 1 is a connected component it results x ∈ U 1 . Setting B 0 := B(y 0 , 2r 3 ), consider the local homeomorphisms f n : Q n → B 0 . Let us check that f n is a proper map, i.e. f −1 n (K) is compact in Q n for every compact set K of B 0 . Indeed, consider a sequence x h ∈ f −1 n (K). By compactness, by passing to subsequences, we may assume that Since we may assume that B 0 is connected, it turns out that (see for instance [8, Th. 1]) f n is onto. Moreover, by [8, Lemma 3] f n is a covering map.
Moreover, since we may assume that B 0 is simply connected, we find (see also [8,Th. 2]) that each f n maps homeomorphically Q n onto B 0 .
As in the previous part of the proof, we find a connected normal neighborhood x 0 ∈ U 2 ⊂ B(x 0 , s) such that f (U 2 ) = B(y 0 , r 3 ). For every x ∈ U := U 1 ∩ U 2 we have Therefore, U ⊂ f −1 n (B 0 ) ∩ U 1 which implies that U ⊂ Q n . Therefore, the restriction of f n to U is injective for every n ≥ 1. Proof. Suppose by contradiction that f (x 1 ) = f (x 2 ) with x 1 = x 2 . Fixed ε > 0, by uniform convergence and the equi-Lipschitz condition we get where H is a common Lipschitz constant for f −1 n . By the arbitrariness of ε we get the contradiction If the sequence f −1 n is uniformly convergent, denote by g n := f −1 n . Hence g n → g uniformly. We derive the contradiction For the local statement, fixed x 0 ∈ X, following the notation of the proof of Lemma 3.8, by reducing the radius r if necessary, we may assume that conditions (i) or (ii) are satisfied on B 0 . Then, we may use the same above arguments on the common neighborhood U .
In the sequel we will provide some more conditions under which the limit of invertible maps is still an invertible map.
For N = 1 the quantity D − f (t) coincide a.e. with the metric derivative |f |(t) (see [5]). Observe that the length of a Lipschitz curve coincides with the total variation (see [5,Th. 4.1.6]). Moreover, the total variation is a lower semicontinuous functional as the supremum of a family of continuous functionals (see the proof of [5,Th. 4

.3.2]).
By using the metric area formula and the l.s.c. of the total variation, using the same reasonings of point (iii) of Section 2, we can state the following Proof. Following the reasonings of point (iii) of Section 2, we get The above computation is related to the lower semicontinuity of the Mass of metric currents. Actually, every Lipschitz map f : Ω → Y induces in a canonical way a 1dimensional metric current (see [2]). See also [18] for an explicit computation. For N = 1 the mass is given just by I |f |(t) dt.
In higher dimensions the question is more complicated since in the computation of the mass a volume factor λ appears.
Precisely, given a norm · on R N , the volume factor λ · (see [2]) is defined as where as usual ω N denotes the volume of the unit sphere in R N . It is possible to show Introducing the metric differential (see [24]) which is defined a.e. and it results a seminorm on R N for a.e. x ∈ Ω, the total mass correspondent to the (invertible) Lipschitz map f (see [3]) is given by where, for a seminorm s on R N , is the metric Jacobian. The metric Jacobian is also related to the metric area formula ( [2,23,28,24]) and for a.e.
x ∈ Ω it can be also expressed by the volume derivative Another way to introduce the metric Jacobian is by considering the so called pullback measure (see [28]). Anyway, for Lipschitz maps all these notions coincide a.e.

3.3.
Lower semicontinuity of metric Jacobians. Given A, B ⊂ Y we define the Haus- We say that A n H → iff d H (A n , A) → 0. Observe that d H is actually a semi-distance, while it is a distance on closed sets. We have the following semicontinuity properties (Lower semicontinuity): Let K ⊂ Y be closed and bounded. If K \ A n H → K \ A, then Proof. Let ε > 0. For large n we have A n ⊂ A ε . Hence it results lim sup n→+∞ H N (A n ) ≤ H N (A ε ). Since H N is non-trivial, by continuity properties of the Hausdorff measure we have By additivity of H N , (9) follows.
Let Ω ⊂ R N be bounded and let f n , f : Ω → Y be normal Lipschitz maps such that f n → f (locally) uniformly. Moreover, suppose that f is discrete. Then, for every open set U ⊂ Ω it results Proof. Observe that by Lemma 3.1 f is open. Fixed ε > 0, we prove that for large n it results If not, we find a sequence y n ∈ f (Ω)\f n (U ) such that d(y n , z) ≥ ε for every z ∈ f (Ω)\f (U ).
Passing to a subsequence, we may suppose y n → y ∈ f (Ω). It results y ∈ f (U ). Let y = f (x 0 ) and let B 0 ⊂⊂ U be a neighborhood of x 0 such that f (x) = f (x 0 ) for every x ∈B 0 . Consider a point z n realizing min z∈fn(B 0 ) d(y n , z).
By the continuity property of balls of Y , we find z n ∈ ∂f n (B 0 ) such that z n → y. Since f n is normal, we get z n = f n (x n ), with x n ∈ ∂B 0 . By uniform convergence and by passing to a subsequence if necessary, we get y = f (x) with x ∈ ∂B 0 , contradicting the discreteness of f . By (9) we get Remark 3.13. Actually, in the proof of the above Lemma it suffices that f is normal and ∂f n (B) → ∂f (B).
Lemma 3.14. Let f n , f : Ω → Y be normal Lipschitz maps such that f n → f (locally) uniformly. Moreover, suppose that f is locally invertible. Then where we denote by Jf (x) := J(mdf (x)) the metric Jacobian of f .

Proof.
Since Ω can be covered by disjoint balls, we may assume with no loss of generality that Ω is bounded. Since f is locally invertible, by a Vitali covering argument, Ω is a disjoint union of balls B i on which f is invertible. By using the metric area formula, Lemma 3.12 and Fatou's Lemma we evaluate Observe that the Jacobian integral is always u.s.c. (for sequences of invertible maps) on closed (or on sets with boundaries of null measure) sets. Indeed, for such a set C, by the Area Formula and (8) In the case Y = R N , the easiest way to obtain l.s.c. of Jacobians is maybe for equi-Lipschitz sequences. In such case, one can use the weak continuity of Jacobians, see for instance [4]. Also in the metric target case, this is true with no invertibility assumptions on f . Proof. Since f is Lipschitz, following the arguments of [24] let D r be the Borel set on which the metric Jacobian exists and it is a norm. It results, up to a null measure set, Jf (x) = 0 for x / ∈ D r . By the results of [24], we find a decomposition of D r on disjoint compact sets E i ⊂ Ω on which f is invertible. Fixed ε > 0, we also find open sets U i such that E i ⊂ U i ⊂ Ω and L N (U i \ E i ) < ε 2 i . We now check the l.s.c. of Jacobians.
In the case Y = R N and if f ∈ C 1 no assumptions on f are actually needed. In fact, in such case we may write Ω = U ∪ B f where B f is the branch set of f , namely the set of points where f is not locally invertible. By Local Inversion Theorem we have B f ⊂ Z f , where Z f := {x ∈ Ω : f differentiable at x, Jf (x) = 0}. Moreover, since f is locally invertible on the open set U , we show that f n : U → R N are open maps for large n.
Indeed, for every x ∈ U , concerning the topological index (see for instance [13,30] for the basic properties) it results i(x, f ) = ±1. Fix a ball B on which f is invertible. For x ∈ B, by uniform convergence, for large n we have Let y = f n (x) ∈ f n (V ) for an open set V ⊂ U . For y ∈ B(y, r n ) in a small ball we get deg(y , f n , B) = deg(y, f n , B) = ±1.
Therefore, there exists x ∈ B such that y = f n (x ). Hence y ∈ B(y, r n ) ⊂ f n (B) ⊂ f n (V ).
Hence, Lemma 3.14 applies. This is also the case of everywhere differentiable maps (see [29]). The same argument also holds for Sobolev maps under topological conditions. Actually, if f is open and discrete, or discrete and sense-preserving, then B f ⊂ Z f ∪ S f , where S f := {x ∈ Ω : f is not differentiable at x} (see [32]). Analogous results in a metric setting could also related to structure properties of the branch set B f .
As consequence of the previous results we get the following stability result.
Theorem 3.16. Let f n , f : Ω → Y (resp. Y be locally pathwise connected and simply connected) be Lipschitz maps such that (resp. locally equi-Lipschitz) Lip(f n ) ≤ M (or more generally such that the Jacobians are equi-integrable) and f n → f (locally) uniformly.
Moreover, suppose that f n are invertible (resp. local homeomorphisms), f (Ω) is open (resp. f is open) and f is discrete. Then f is invertible (resp. locally).
Proof. By Lemma 3.3 f is an open map. Suppose with no loss of generality that Ω is bounded. By using Theorem 3.15 it is sufficient to argue as in Section 2. Actually, suppose that f (x 1 ) = f (x 2 ) with x 1 = x 2 . We find small disjoint balls is a non-empty open set. We compute by (8) and since H N (∂V ) = 0. For the local statement it is sufficient to apply Lemma 3.8. Theorem 3.16 can be compared with the results of [31,25,26]. A main point here is that we are not using degree theory.

Small contraction maps
Maps satisfying (11) 0 were introduced by John (see for instance [22]). We may obtain some results related to quasi-isometries by using some of the above arguments. Let us mention some preliminary basic facts (see [12,7,22]).
Definition 4.1. A metric space E is said to be C-convex if for every a, b ∈ E there exists a rectifiable curve γ joining a, b such that length(γ) ≤ Cd(a, b).
It turns out that C-convex spaces are bi-Lipschitz homeomorphic to a length metric space.  Proof. By Remark 2.5 and Lemma 3.3 of [19] it follows D + f −1 (f (x)) = 1 D − f (x) . By Lemma 4.2 it follows that the sequence f −1 n is equi-Lipschitz. By using the second part (ii) of Lemma 3.9 the thesis follows.  In order to get inversion results, we consider the following variant of Ciarlet-Necas condition (12) for open set U ⊂ X : m N L N (U ) ≤ H N (f (U )).
By using the area formula, condition (12) is trivially satisfied by invertible maps such that Jf ≥ m N . This inversion results can be compared with the results of [22,15,16,17]. Actually, by (12) we compute H N (f (B(x, r))) ω N r N ≥ m N .
Passing to the limit we obtain Jf (x) ≥ m N . Observe that our metric approach allow to consider a general open set Ω ⊂ R N , while the results of [22,15,16,17] are restricted to particular geometries of the domain of f . In particular, maps satisfying Jf ≥ m N are considered in [17] in the case of local homeomorphisms f : B → R N , defined on a ball B ⊂ R N . Therefore, the restriction to balls or to locally invertible maps could be not strictly necessary. Another advantage of the volume condition (12) is the stability with respect to uniform convergence.
Theorem 4.7. Let f n : Ω → Y be M -Lipschitz maps satisfying (12). Suppose that f n → f uniformly. If f is open and M m < N √ 2, then f is invertible.