Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values

Calc. Var. (2026) 65:200 https://doi.org/10.1007/s00526-026-03370-w Calculus of Variations Global invertibility of Sobolev mappings with prescribed homeomorphic boundary values Sabrina Traver1 Received: 6 July 2025 / Accepted: 26 May 2026 © The Author(s) 2026 Abstract Let X , Y ⊂ Rn be Lipschitz domains, and suppose there is a homeomorphism ϕ : X → Y . We consider the class of Sobolev mappings f ∈ W 1,n (X , Rn ) with a strictly positive Jacobian determinant almost everywhere, whose Sobolev trace coincides with ϕ on ∂ X . We prove that every mapping in this class extends continuously to X and is a monotone (continuous) surjection from X onto Y in the sense of C. B. Morrey. As monotone mappings, they may squeeze but not fold the reference configuration X . This behavior reflects weak interpenetration of matter, as opposed to folding, which corresponds to strong interpenetration. Despite allowing weak interpenetration of matter, these maps are globally invertible, generalizing the pioneering work of J.M. Ball. Mathematics Subject Classification 46E35 1 Introduction Throughout this paper, X and Y will represent bounded Lipschitz domains in Rn . Geometric Function Theory, particularly within variational frameworks [1–4], and the theory of Nonlinear Elasticity [3, 5, 6] studies elastic bodies X and their deformations f : X → Rn . Properties of these deformations correspond to physical phenomena, making some classes of mappings better suited to the study of materials science. Most notably, wellbehaved deformations are injective (or at least injective a.e.) because injectivity is essential for ensuring non-interpenetration of matter. However, even C 1 -mappings f with strictly positive Jacobian determinant J f (x) = det D f (x) > 0, which are locally injective by the classical inverse function theorem, may fail to be injective on the whole domain. This failure may arise, for example, when different ends of the domain are mapped to overlapping subsets of the target. The result is selfinterpenetration of matter, but it can be shown using Degree Theory that when the C 1 map f coincides with an orientation-preserving homeomorphism ϕ : X → Y on the boundary Communicated by A. Mondino. B 1 Sabrina Traver Department of Mathematics, Syracuse University, 215 Carnegie, Syracuse 13244, New York, United States 0123456789().: V,-vol 123 200 Page 2 of 14 S. Traver of the domain X , this cannot occur. More precisely, C 1 mappings f : X → Rn with strictly positive Jacobian determinant are homeomorphisms from X onto Y when f |∂ X = ϕ|∂ X . Replacing the C 1 maps, f and ϕ, with Sobolev maps, injectivity is generally lost (see example 3), but a weaker version of injectivity (injectivity a.e.) is still maintained. Moreover, mappings that are injective a.e., also known as globally invertible, are sufficient for physical applications [7–9]. We say a map f : X → Y is globally invertible if |{y ∈ Y : f −1 (y) is a single point}| = |Y | (1) Equation (1) is often referred to as global invertibility [7] or invertibility for a.e. y ∈ Y because when (1) holds, then a.e. y ∈ Y has a unique preimage. Each map in the following class of deformations 1, p Aϕp = { f ∈ W 1, p (X , Rn ) : J f > 0 a.e., f − ϕ ∈ W0 (X , Rn )}. is globally invertible for p ≥ n. The p > n case can be found in [7], and, as we will show in Section 6 the p = n case follows from Theorem 1. When p = n such maps are also monotone in the sense of Morrey [10]: Definition 1 Given compact, connected metric spaces, X and Y , a continuous map f : X → Y is monotone if f −1 (y0 ) is connected for all y0 ∈ Y . Monotonicity allows for weak interpenetration of matter. That is, roughly speaking, squeezing a portion of the material to a point can occur but not folding or tearing. We now introduce our main result: Theorem 1 Let X and Y be bounded Lipschitz domains in Rn and ϕ : X → Y be a given homeomorphism in W 1,n (X , Rn ). If f ∈ W 1,n (X , Rn ) with J f > 0 a.e. such that f − ϕ ∈ W01,n (X , Y ) then 1. 2. 3. 4. f f f f has a continuous extension to the boundary of X , which we still denote f maps X onto Y : X → Y is a monotone map : X → Y is globally invertible Corollary 1 Let f be as in Theorem 1. Then there is a set X f ⊂ X of full measure such that f , when restricted to X f , is injective. The W 1, p version of Theorem 1 for p > n was proven by Ball in 1981 [7]. Though the results are similar, the conformally invariant case p = n has one significant difference: for p > n, the boundary regularity follows directly from continuity estimates for arbitrary Sobolev maps. In the critical case p = n, such estimates break down, so we devise new estimates by analyzing oscillation and applying a Sobolev embedding theorem. Even in the p = n case, slight changes to the class of maps may produce different results. As an example, Ball, Iwaniec, and Onninen [11] previously studied weakly convergent sequences of W 1,n −Sobolev homeomorphisms in the traction free setting. When boundary data is not fixed, they showed that uniform equicontinuity may fail without additional topological or variational assumptions [11]. In contrast, the present paper relies on homeomorphic boundary data to prove many properties of the map in Theorem 1, such as continuity up to the boundary and global invertibility. In section 6, we apply our result to solve an energy minimization problem. Although the classical results of Ioffe [12] could potentially be applied to this and similar settings, the singular behavior of the integrand in combination with the assumption J f > 0 a.e. means 123 Global invertibility of Sobolev mappings… Page 3 of 14 200 a direct proof is better suited to demonstrate the unique features of this functional. Indeed, the proof relies on convexity, weak L 1loc convergence of Jacobians, and an application of the Monotone Convergence Theorem. The last section, 7, is reserved for examples. In particular, we show the strict inequality J f > 0 a.e. is essential. In Example 1, if we allow J f ≥ 0, Theorem 1(1) and (3) may fail, and even if f is also continuous up to the boundary, as in Example 2, Theorem 1(2) may fail. 2 Preliminaries The domains in this preliminaries section need not be Lipschitz. We say X is a Lipschitz domain if for each x ∈ ∂ X there exists a neighborhood Ux of x and a Cartesian coordinate system x̂ = {x1 , . . . , xn } in Ux such that X ∩ Ux = {x̂ ∈ Ux : xn > f (x1 , . . . , xn−1 )} for some Lipschitz continuous function f : Rn−1 → R. We begin by introducing the Lusin N condition, which, when considering physical applications, ensures that our deformation f cannot create material from nothing. Mathematically, for instance, its value is in its ability to relate the Jacobian of our map f to a degree function, which counts the number of preimages under f of a given point, while accounting for orientation. Definition 2 Let X ⊂ Rn be open. A map f : X → Rn satisfies the Lusin N condition if for every E ⊂ X with |E| = 0, | f (...truncated)