SOME REMARKS ON SOBOLEV AND BI-SOBOLEV MAPPINGS

Journal of Mathematical Sciences (2024) 280:50–60 https://doi.org/10.1007/s10958-023-06665-x SOME REMARKS ON SOBOLEV AND BI‑SOBOLEV MAPPINGS Luigi D’Onofrio1 · Giovanni Molica Bisci2 Accepted: 25 September 2023 / Published online: 21 October 2023 © The Author(s) 2023 Abstract In this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev 1,1 homeomorphism if f belongs to Wloc ∩ Hom(Ω;Ω� ), and f is a bi-Sobolev map if and only if f and f −1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268 2017). Keywords Sobolev maps · BV homeomoprshims MSC Classification Primary 46E35 · Secondary 26B30 Introduction Let us consider a planar homeomorphism f ∶ Ω → Ω� , where we set Ω� ∶= f (Ω) . By using the trivial equality |M| = det M|M −1 | for any 2 × 2 invertible matrix M, if the change variables formula holds, then we can formally calculate (see [30]) ∫ Ω� |Df −1 (y)|dy = ∫Ω |Df −1 (f (x))Jf (x)|dx = ∫Ω |Df (x)|dx (1.1) where with Df, we mean the differential matrix for the map f and Jf its Jacobian determinant. There are two problems that naturally appear in the above computations. First of all, it is well known that the change of variable formula is true only if the Lusin property holds for the map f; in the literature there are examples that show that this is not true for bi-Sobolev mappings (see [28]). Just for the reader, we recall some notations and definitions: Giovanni Molica Bisci contributed equally to this work. * Luigi D’Onofrio Giovanni Molica Bisci 1 Dipartimento di Scienze e Tecnologie, Universita’ degli Studi di Napoli “Parthenope”, Napoli 80100, Italy 2 Dipartimento di Scienze Pure e Applicate, Universita’ degli Studi di Urbino Carlo Bo’, Urbino 10587, Italy Vol.:(0123456789) 51 Journal of Mathematical Sciences (2024) 280:50–60 onto • we will write f ∈ Hom(Ω, Ω� ) if and only if f ∶ Ω ⟶ Ω� is a homeomorphism. • f satisfies the Lusin condition (N) if (1.2) for every measurable E ⊂ Ω with |E| = 0 we have |f (E)| = 0. During this paper | ⋅ | denotes either the standard Lebesgue measure or the norm of the differential matrix; however, its meaning will be clear from the context. 1,1 • f is a Sobolev homeomorphism if f −1 belongs to Wloc ∩ Hom(Ω� ;Ω), in such a case f is a bi-Sobolev map, i.e., if f and −1 are Sobolev homeomorphisms [10]. f Secondly, the formal calculation in Eq. 1.1 requires that both f and f −1 satisfy the Lusin (N)-condition. Indeed, in Eq. 1.1, we use the change of variable and the composition between f and its inverse, these facts are allowed if and only if f, f −1 satisfy the Lusin (N)- condition, indeed a bi-Sobolev map does not satisfy the Lusin condition [10]. However, the bi-Sobolev mappings satisfy an interesting property, that is, in the plane a bi-Sobolev mapping is a map of finite distortion, i.e., for almost every x ∈ Ω for which Jf (x) = 0 one actually has Df (x) = 0. This technical fact plays a key role in the study of bi-Sobolev homeomorphisms. We have the following (see [18, 21]) Theorem 1.1 Let f ∶ Ω → Ω� be a bi-Sobolev mapping. Then, ∫Ω |Df (z)| dz = ∫Ω� (1.3) |Df −1 (w)| dw. The bi-Sobolev assumption rules out a large class of homeomorphisms. For example, the mapping onto f0 ∶ (0, 2) × (0, 1)⟶(0, 1) × (0, 1) (1.4) f0 (x, y) = (h(x), y) where h−1 (t) = t + c(t) and c ∶ (0, 1) → (0, 1) is the usual Cantor ternary function, h is a Lipschitz homeomorphism whose 1,1 inverse f0−1 does not belong to Wloc . However, for this kind of mappings analogous identities have been recently found in [7, 9, 13]. The main scopes of this short note are as follows: • to clarify the assumption that guarantees the validity of the change variable formula (the Lusin (N)-condition); • to identify the results and counterexamples for mappings that failed to be bi-Sobolev The paper is organized as follows: In “Lusin condition and area formula,” we discuss the Lusin condition to answer to the first question above. In “Sobolev and BV-homeomorphisms,” we introduce the concept of bi-Sobolev mappings to discuss the second question above. The final section is devoted to sense preserving homeomorpshim in the plane. These assumptions are crucial to give a full characteritazion of the total variation of the inverse. Lusin condition and area formula onto Let Ω ⊂ ℝn be a bounded domain of the Euclidean space ℝn . For a homeomorphism f ∶ Ω ⟶ Ω�, condition Eq. 1.2 holds if and only if the function f maps measurable sets to measurable sets. Moreover, if f is differentiable at every point x ∈ B, where B ⊂ Ω is a Borel set, and Jf (x) is the Jacobian determinant of f at x ∈ B, then the weak area formula (see Federer [15, Theorem 3.1.8]) holds on B, that is, �B 𝜂(f (z))Jf (z)dz ≤ �f (B) 𝜂(w)dw, (2.5) for any nonnegative Borel-measurable function 𝜂 ∶ ℝn → [0, +∞[. However, the (N) condition for such f on B ⊂ Ω is equivalent to the area formula [25] 52 Journal of Mathematical Sciences (2024) 280:50–60 ∫B 𝜂(f (z))|Jf (z)|dz = ∫f (B) 𝜂(w)dw. (2.6) 1,n If the homeomorphism f satisfies the natural assumption f ∈ Wloc (Ω;ℝn ), then f verifies the Lusin (N)-condition given in Eq. 1.2. This result due to Reshetnjak [31] is sharp in the scale of W 1,p (Ω;ℝn )-homeomorphisms thanks to a quoted example of a W 1,p-homeomorphism f ∶ [0, 1]n → [0, 1]n , with p < n, for which condition (N) fails; see Ponomarev [28, 29] for details. For the sake of completeness, let us recall some auxiliary results (see, for example, [12]). Proposition 2.1 Let f ∈ Hom(Ω;Ω� ). Then, f satisfies the Lusin (N)- condition iff f preserves the measurability property, i.e., f(E) is measurable, for every measurable set E ⊆ Ω. Proposition 2.2 A homeomorphism f ∶ Ω → Ω� satisfies the condition (N) iff |f (E)| = 0 whenever E ⊂⊂ Ω is a compact set with zero measure. Proof If E ⊂ Ω satisfies |E| = 0, then there exists a Borel set B ⊃ E such that |B| = 0. By contradiction if |f (B)| > 0, there exists a compact set C� ⊂ f (B) such that |C′ | > 0. On the other hand, since f is a homeomorphism, f −1 (C� ) is compact and |f −1 (C� )| ≤ |B| = 0. This is not possible by assumption.  ◻ The weak area formula follows from the area formula Eq. 2.6 which is valid for Lipschitz mappings and from the fact that the set of differentiability can be exhausted up to a set of zero measure by sets restricted to which f is Liponto schitz ([15] Theorem 3.1.8]). Hence, for an almost everywhere differentiable homeomorphism f ∶ Ω ⟶ Ω� , we can decompose Ω into pairwise disjoint sets Ω=Z∪ ∞ ⋃ Ωk k=1 (2.7) such that |Z| = 0 and f |Ωk is a Lipschitz continuous function, for every k ∈ ℕ. We emphasize the following (...truncated)