Manifold-constrained free discontinuity problems and Sobolev approximation

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset \R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.


Introduction
Let Ω ⊂ R 2 be a bounded open set, p : Ω → [1, +∞) be a measurable function (that will be called a variable exponent in the following course), and M be a compact, connected, smooth Riemannian manifold without boundary. In this paper, we deal with special functions of bounded variations from Ω into M whose approximate differential is integrable with respect to the variable exponent p(·) over Ω and whose jump set has finite H 1 -measure. The space of such functions will be denoted by the symbol SBV p(·) (Ω, M); we refer the reader to Section 2 for its formal definition and as well as for basic material on functions that are integrable with respect to variable exponents.
In this work, we first prove an approximation result for maps of class SBV p(·) (B ρ , M) with small jump set by functions that are also of Sobolev class in slightly smaller balls. (Here, B ρ denotes any ball in R 2 .) Then, we apply such a result to the study of the regularity of local minimisers among S k−1 -valued maps, where S k−1 denotes the unit sphere in R k and k ≥ 2, of an energy functional with nonstandard growth. More precisely, we consider, for u ∈ SBV Ω, S k−1 , the following energy functional: The functional F (u, Ω) is finite exactly on SBV p(·) Ω, S k−1 and and we say that a function u belonging to SBV p(·) Ω, S k−1 is a local minimiser (among S k−1 -valued maps) of F if and only if F (u, Ω) ≤ F (v, Ω) for every v ∈ SBV Ω, S k−1 such that {u = v} ⊂⊂ Ω. The functional F may be seen as a prototypical energy involving both a manifold-valued constraint on the maps and a nonstandard growth (more precisely, p(·)-growth) in the energy functional. Note that the functional F reduces to the p(·)-energy w →ˆΩ \Ju |∇w| p(x) dx out of the closure of the jump set J u of u. Since bounded special functions of bounded variation are of Sobolev class outside J u , we see that any local minimiser of F is a local minimiser among S k−1 -valued maps of the p(·)-energy (i.e., a p(·)-harmonic map into S k−1 ) in the open set Ω \ J u . However, although J u is always negligible for the Lebesgue measure, there is no reason, a priori, for J u to not be the whole Ω and, as we shall see in the next paragraphs, proving that J u is "small" (more precisely, essentially closed) is indeed the most difficult step towards the regularity of local minimisers of F .

Main results
In order to state our main results, let us set the basic notation, referring to Section 2 for more terminology. In all this work, unless stated otherwise, the following assumptions on the variable exponent p : Ω → [1, +∞) will be in force: p is log-Hölder continuous; for all x ∈ Ω, 1 < p − ≤ p(x) ≤ p + < 2, where p − := ess inf x∈Ω p(x) and p + := ess sup x∈Ω p(x). The assumption of log-Hölder continuity is customary in the study of functionals with p(·)-growth, c.f., eg., [40,1]. We recall this notion along with its geometric meaning in Definition 2.1 and in Lemma 2.4 below. (We are going to comment on the assumption p + < 2 in the next paragraphs.) On the manifold M, we assume throughout it is a compact, connected, smooth Riemannian manifold without boundary, isometrically embedded in R k , for some k ∈ N. Under these assumptions, it well-known that there exists a locally smooth retraction P : R k \ X → M, where X is a smooth complex of codimension 2, with locally q-integrable gradient, for every q ∈ [1,2). (See Section 3.2 for more details.) This retraction plays a key rôle in the proof of our first main result, Theorem A below.
Theorem A. Let p : Ω → (1, +∞) be a variable exponent satisfying (p 1 ), (p 2 ). Let M be a compact, connected, smooth Riemannian manifold without boundary, isometrically embedded in R k , for some k ∈ N. There exist universal constants ξ, η > 0 such that for any s ∈ (0, 1) and any u ∈ SBV p(·) (B ρ , M) (where B ρ is any ball in R 2 ) satisfying there exists a function w ∈ W 1,p(·) (B sρ , M) ∩ SBV p(·) (B ρ , M) and a family F of balls such that the following holds. The function w coincides with u a.e. outside of the union of the balls in the family F. Such a union is contained in B (1+s)ρ/2 and controlled in measure and perimeter by ξ, η, ρ, and H 1 (J u ∩ B ρ ). Moreover, w has less jump than u in B ρ and´B ρ |∇w| p(x) dx is controlled by ρ, p − , p + , the log-Hölder constant of p(·), M, P, k, and ∇u L p(·) (Bρ) .
Theorem A is an abridged version of a more precise and descriptive statement, Theorem 3.1 below. Theorem A may be seen as an extension to our manifold-valued, variable-exponent setting of a result by Conti, Focardi, and Iurlano [16], which was developed in the context of planar domains, constant exponents, and (unconstrained) SBD p functions (i.e., for special functions of bounded deformation with approximate symmetric gradient in L p and jump set with finite H 1 -measure) with small jump set, and later applied in [17] to the study of Griffith's type brittle fracture functionals. As we shall see in more detail later, the result is confined to the two dimensional setting and cannot be directly extended to higher dimensions.
The seminal idea for studying the regularity of local minimisers of free discontinuity problems for maps of class SBV is due to De Giorgi, Carriero, and Leaci [25] and lies in showing that the jump set of any local minimiser u is essentially closed, i.e., satisfies In the constant exponent case, once this is done, standard elliptic regularity yields that These ideas, firstly developed for scalar-valued functions in [25], were extended to the case of S k−1 -valued maps (and constant p) in [11]. Very recently, they have been adapted to the scalarvalued, variable-exponent setting in [34] (for a larger class of convex functionals, of which (1) is the prototype). In all these works, a major technical tool is the Sobolev-Poincaré inequality for SBV-functions due, once again, to De Giorgi, Carriero, and Leaci [25]. Here, we follow the approach devised by Conti, Focardi, and Iurlano [16,17], which avoids the use of truncations (fundamental in the proof of the Sobolev-Poincaré inequality in [25]), by relying, instead, on the Sobolev approximation. With the aid of Theorem A, we prove Theorem B below, which is our second main result in this paper. Before stating the theorem, we have to introduce a strengthening on the assumption (p 1 ): p is strongly log-Hölder continuous, (p ′ 1 ) In the calculus of variations, energy functionals with p(·)-growth in the gradient have been proposed in the modelling of materials which exhibit a strongly anisotropic behaviour starting from the works [38,39]. In the Sobolev setting, the regularity of minimisers has been analysed to a certain degree of generality, both in the scalar and in the vector-valued, unconstrained case, see, e.g., [19,1]. Less literature is available for manifold-constrained maps. However, the regularity problem for p(·)-harmonic maps has been recently considered in [23,15].
In the last years, several works [22,3,36,34] have undertaken the study of energy functionals with nonstandard growth defined over spaces of maps of class SBV p(·) or even GSBV p(·) or GSBV ψ . The aims of these works range from lower semicontinuity results [22,3] to integral representation theorems [36], up to regularity in the scalar-valued case [34].
Spaces of functions of (special) bounded variation and values into a Riemannian manifold M have been recently studied in [33,9]. The authors of [33,9] are mainly concerned with constructing liftings of such mappings from the manifold M to its universal cover M. The interest towards such problem was initially stimulated by applications to the Landau-de Gennes theory of liquid crystals and to the Ginzburg-Landau model of superconductivity.
The present work is, to the best of our knowledge, the first that considers the prototypical energy functional F in (1) for manifold-valued special functions of bounded variation with L p(·) -integrable approximate differential. The functional F in (1) is itself a particular case of a more general one: where u ∈ SBV p(x) (Ω, M) and φ 0 is BV-elliptic, see [4,Definition 5.13], and bounded away from zero. Functionals like these appear in the theory of liquid crystals, for example nematics and, in this case, the manifold is isomorphic to S 1 . Another model, related to smectic thin films, can be found in a recent paper [5]. Here a free discontinuity problem is proposed in order to describe surface defects in a smectic thin film. The free energy functional contains an interfacial energy, which penalises dislocations of the smectic layers at the jump. The function space is a subspace of function of bounded variation SBV 2 (Ω, R 2 ) with values in a suitable manifold N.

Proofs of the main results: a sketch
The proof of Theorem A (better, of Theorem 3.1) is contained in Section 3. Essentially, it proceeds in two steps: first, we prove an analogous statement for unconstrained maps with values into R k (Proposition 3.8), following very closely the original argument in [16, Theorem 2.1 and Proposition 3.2] and exploiting the assumption of log-Hölder continuity of the variable exponent. Then, we use the aforementioned retraction P to obtain Theorem A from its unconstrained counterpart, by retraction onto the manifold of the image of the unconstrained approximating maps. The restriction on the dimension of domain in Theorem A comes from the fact that, to craft the Sobolev approximation, we use the same construction as in [16], which is strictly two dimensional and cannot be extended (without heavy modifications) to higher dimensions (see [7] and Remark 3.6 and Appendix C below for more details on this point). The restriction p + < 2 in (p 2 ) is due instead to the usage of the retraction P in the proof of Theorem A along with the fact that we merely require connectedness on M (so to allow for M to be a circle, an important case in potential applications -for instance, to liquid crystals), see Lemma 3.11 and Remark 3.13 for more details. However, since we work in dimension 2, such a restriction is not a dramatic drawback, in the sense that this is the subcritical regime for Sobolev-Morrey's embedding and maps of class W 1,p(·) (Ω, M) are not automatically continuous (neither in Ω nor in open subsets of Ω with positive measure). In other words, this is the regime in which all the essential complications in the study of the regularity of local minimisers of the functional F in (1) already show up, without the additional ones due to possibly large oscillations of the variable exponent.
As alluded few lines above, the assumption of log-Hölder continuity of the variable exponent is extremely important in the proof of Theorem A and this can be easily realised by looking at its geometric meaning (see Lemma 2.4). Indeed, roughly speaking, such an assumption boils down to the possibility of locally "freezing" the variable exponent, up to a controlled error. Joint with the very precise estimates for the approximating Sobolev map coming from [16], this yields a rather direct extension of the results of [16] to our variable-exponent setting, at least in the unconstrained case (compare the proofs of Proposition 3.3 and Proposition 3.8 with those of [16, Theorem 2.1 and Proposition 3.2]).
The proof of Theorem B proceeds in various steps, following the pioneering approach from [25,11]. The key point relies in proving a suitable power-decay of the energy in small balls with the radius of the ball. This is done in Theorem 4.3 by means of a classical argument by contradiction and a blow-up analysis, relying on assumption (p ′ 1 ). We adapt the strategy of [17] to exploit, in the blow-up analysis, Theorem A (with M = S k−1 and k ≥ 2) instead of the classical Sobolev-Poincaré inequality in SBV (adapted to the variable-exponent framework in [36,34]).
Once the decay lemma is obtained, another classical argument originating from [25] yields suitable density lower bounds for F (u, B ρ (x)), where u is a local minimiser of F , x ∈ J u , and ρ is small. In turn, such density lower bounds readily imply, by a standard argument in geometric measure theory, that J u is essentially closed. This step requires more than log-Hölder continuity and indeed we ask strong log-Hölder continuity in the statement of Theorem B. To the purpose of proving essential closedness of the jump set, strong log-Hölder continuity turns out to be enough and, actually, also local minimality can be weakened to quasi-minimality (see Definition 4.2). The full strength of Hölder continuity and of local minimality are instead needed to prove that u ∈ C 1,β 0 where Ω 0 is as in the statement of Theorem B. Indeed, here we use the regularity result [23, Theorem 1] (in the simplified form provided by Theorem 2.8 below) for the p(·)-harmonic energy for manifold-valued maps, i.e., for local minimisers (among compactly supported perturbations) of the functional which requires p ∈ C 0,α (Ω), for some α ∈ (0, 1], among its assumptions.
Organisation of the paper In Section 2, we establish notation and recall the basics facts about spaces of functions integrable with respect to variable exponents. In Section 3, we prove Theorem A. In Section 4, we prove Theorem B. The paper is completed by a series of appendices containing mostly technical material for which we were not able to find explicit proof in the literature. In Appendix C, we exhibit an example which shows that the approximation procedure in [16] and, in turn, in Section 3 cannot work in higher dimensional domains under the mere assumption of H 1 -smallness of the jump set.

Notation
(i) We use the symbol {x n } to denote a sequence, indexed by n ∈ N, of elements x n of a certain set E. Usually, for the sake of a lighter notation, we do not relabel subsequences.
(ii) In inequalities like A B, the symbol means that there exists a constant C, independent of A and B, such that A ≤ CB.
(iii) We denote B n ρ (x 0 ) := {x ∈ R n : |x − x 0 | < ρ} the open ball of radius ρ and centre x 0 in R n . Since we work almost exclusively in dimension n = 2, we drop the superscript "2" for balls in R 2 . Often, the centre of the ball will be irrelevant and we shall omit it from the notation. Given a ball B ρ , the symbol B sρ denotes the ball concentric with B ρ and radius dilated by the factor s > 0.
(iv) We denote by M a compact, connected, smooth Riemannian manifold without boundary, of dimension m ≥ 1. Without loss of generality, we may always view at M as a compact, connected, m-dimensional smooth submanifold in R k , for some k ∈ N. This can always be achieved, for instance, by means of Nash's isometric embedding theorem. If not specified otherwise, we will always assume to embed M in R k via Nash's embedding. By compactness of M, we can find M > 0, depending only on M and the choice of the isometric embedding, such that In the following course, when saying that a quantity "depends on M", we will always mean that it depends on M and the chosen isometric embedding of M into R k . However, by compactness of M, the choice of the isometric embedding is essentially irrelevant, in the sense that changing the embedding can result, at worst, in enlarging k and the constants that depend on M.
(v) In the special case M is a sphere, we always look at it as a submanifold of R m+1 the obvious way. In particular, we will denote S k−1 t := x ∈ R k : |x| = t the (k − 1)-dimensional sphere in R k of radius t > 0 and centre the origin, endowed with the canonical metric. We set S k−1 := S k−1 1 . The canonical orthonormal basis of R k will be denoted {e 1 , . . . , e k }.
(vi) For any δ > 0, we will denote the portion of the δ-neighborhood of ∂Ω that lies in the interior of Ω.

Variable exponent Lebesgue
In this section, we recall some basic facts about variable-exponent Lebesgue spaces. The reader can consult the monographs [27,21] for more details.
A measurable function p : Ω → [1, +∞) will be called a variable exponent. For every A ⊂ Ω (measurable and not empty) we define Of course, we can take A = Ω, and in this case we write p + and p − in place of p + Ω and p − Ω , respectively. We say that a variable exponent p is bounded if p + < +∞.
The modular ̺ p(·) (u) of a measurable function u : Ω → R m with respect to the variable exponent p(·) is defined by and the Luxembourg norm (henceforth, simply norm) of u by The variable exponent Lebesgue space L p(·) Ω, R k is defined as the set of measurable functions u : Ω → R k such that there exists λ > 0 so that ̺ p(·) (u/λ) < +∞.
We collect the properties of variable exponent Lebesgue space that we will use in the following propositions. Proposition 2.1. Let p : Ω → [+1, ∞) be a variable exponent and assume that p + < +∞. Then: where p ′ is the variable exponent satisfying 1 p + 1 p ′ = 1 a.e. in Ω. (iv) The following inequalities hold: (iii) If f h → f L n -almost everywhere in Ω as h → +∞ and there exists g ∈ L p(·) Ω, R k such that |f h | ≤ |g| a.e., then f h → f in L p(·) Ω, R k as h → +∞.
The following proposition extends to the variable exponent setting the classical embedding property of Lebesgue spaces on sets with finite (Lebesgue) measure.

Remark 2.5.
A bounded, log-Hölder continuous variable exponent defined on a bounded set Ω ⊂ R n can always be extended to a bounded, log-Hölder continuous variable exponent q which is defined on the whole of R n and that satisfies C q ≤ (p − ) 2 C p , ℓ q ≤ (p − ) ℓ, q − = p − , q + = p + , and (2.5) for any open ball B ⊂ R n see, e.g., [27,Proposition 4.1.7]. For our purposes in this paper, there is no loss of generality in assuming from the very beginning that p is defined on the whole R n .
Later in this work we will need the following strengthening of the log-Hölder condition (2.4).

Definition 2.2.
We say that a variable-exponent p(·) : Ω → [1, +∞) satisfies the strong log-Hölder condition in Ω if and only if its modulus of continuity ω satisfies Remark 2.6. The extension to the whole R n of a strongly log-Hölder continuous variable exponent is still strongly log-Hölder continuous.

Variable exponent Sobolev spaces and p(·)-harmonic maps
Let Ω ⊂ R n be an open set and p(·) : R n → R be a variable exponent. As in the classical case, we define turns W 1,p(·) Ω, R k into a Banach space, separable if p + < +∞ and reflexive if p − > 1 and p + < +∞. We address the reader to [27,21] for full details about variable exponent Sobolev spaces.
As in the classical case, we define
We observe that, although [23,Lemma 4] (which in turn is partly borrowed from [32,Lemma 4.5]) is stated assuming j ≥ 1 (i.e., M simply connected), it holds as well for j = 0 (c.f., e.g., [31,6,8] and Lemma 3.11 below). This fact allows us to construct maps with all the properties required by [23,Lemma 5] without assuming M simply connected, c.f., e.g. Lemma 3.16 below. Besides [23,Lemma 4 and Lemma 5], the assumption that M is simply connected is never used in the proof of [23, Theorem 1] (and indeed removed in the follow up [15] of [23]). Consequently, [23, Theorem 1] holds as well even if M is merely connected, provided that p + < 2. In turn, Theorem 2.8 holds if M is merely connected, provided that p + < 2, which are precisely the assumptions under which we work in this paper.

The space SBV p(·) (Ω, M)
Here, we collect some basic facts about SBV functions that are used throughout this paper and we define precisely the space of mappings of class SBV p(·) from an open set Ω ⊂ R n into a Riemannian manifold M, where p(·) is a bounded, variable exponent. We address the reader to [4,Chapters 2,3,4] for details about SBV (and BV) functions as well as for all the notions from Geometric Measure Theory ([4, Chapter 2]) that we will use in this work.
Let Ω ⊂ R n be an open set and k ∈ N. We recall that the set SBV Ω, R k is the linear subspace of L 1 Ω, R k made up by those functions u whose distributional gradient Du is a bounded Radon measure with no Cantor part. Endowed with the norm u BV(Ω) := u L 1 (Ω) + |Du| (Ω), the space SBV Ω, R k is a Banach space. For any u ∈ SBV Ω, R k , its distributional gradient Du splits as Du = (∇u)L 2 where ∇u denotes the approximate differential of u, u ± the traces of u on the two sides of the jump set J u , and ν u is the normal field to J u (see, e. g., [4,Definition 3.67]). We recall from [4, Proposition 3.69 and Theorem 3.78] that J u is a countably H n−1 -rectifiable Borel set in Ω.
Let p ≥ 1 and Endowed with the restriction of the norm, SBV p Ω, R k is a Banach subspace of SBV Ω, R k that proved useful in handling free discontinuity problems at least since [11]. On the relation between the spaces SBV, SBV p , W 1,1 , W 1,p , we recall that: for any open set Ω ⊂ R n , u ∈ W 1,1 Ω, R k ⇐⇒ u ∈ SBV Ω, R k and H n−1 (J u ) = 0.
If Ω is a bounded open set with Lipschitz boundary, then, by iterated application of Sobolev embedding, A straightforward generalisation of the space SBV p Ω, R k has been introduced in [22], by defining the space SBV p(·) Ω, R k , where p(·) : Ω → [1, +∞) is any bounded variable exponent, as the subspace of those u ∈ SBV Ω, R k whose jump set has finite H n−1 -measure and whose approximate differential is L p(·) -integrable on Ω.
In this work, we are mostly interested in maps taking values into a compact Riemannian manifold without boundary. By viewing M as a submanifold of R k , for some k ∈ N (for instance, by means of Nash's isometric embedding theorem), we can define This space enjoys closure and compactness properties analogous to those of the classical space SBV p Ω, R k , see Appendix A for more details.
For later use, we notice that if Ω is any bounded open set, then by compactness of M (which implies u ∈ L ∞ Ω, R k ). Moreover, relaxing the assumption of boundedness of the functions, but assuming instead that Ω is a bounded open set with sufficiently nice boundary (Lipschitz is enough) and that p − < n and p + < p − * , Proof. The direction ⇒ is obvious. For the direction ⇐, we argue as in the classical case of constant exponents, exploiting assumption (2.11): by assumption and (2.8), u ∈ W 1,1 Ω, R k , which implies, by Sobolev embedding, u ∈ L 1 * Ω, R k . If 1 * ≥ p − , then u ∈ W 1,p − Ω, R k , which implies (in view of (2.11)) u ∈ L p + Ω, R k , hence u ∈ L p(·) Ω, R k by Proposition 2.3. Since we know by assumption that ∇u ∈ L p(·) Ω, R k×n , it follows that u ∈ W 1,p(·) Ω, R k . If instead 1 * < p − , then u ∈ W 1,1 * Ω, R k , and we can repeat the above step. After finitely many iterations, we obtain 1 * * ··· * ≥ p − and we conclude as in the above.

Sobolev approximation of functions in SBV p(·) (Ω, M) with small jump set
In this section, we prove Theorem 3.1 below, which clearly entails Theorem A.
be a bounded, log-Hölder-continuous, variable exponent satisfying p − > 1 and p + < 2. Let k ∈ N and ρ > 0. Assume M is a connected, compact Riemannian manifold without boundary, isometrically embedded into R k . There exist universal constants ξ, η > 0 such that for any s ∈ (0, 1) and any u ∈ SBV p(·) (B ρ , M) satisfying the following holds. There are a countable family F = {B} of closed balls, overlapping at most ξ times, of radius r B < (1 − s)ρ and centre x B ∈ B sρ , and a function w ∈ SBV p(·) (B ρ , M) such that where the implicit constant in the right hand side is independent of u and w and depends only on the quantities listed in Remark 3.17 and on the log-Hölder constant of p(·). Moreover, if p is constant, then (3.3) holds without the factor (1 + ρ 2 ) at right hand side.
The proof of Theorem 3.1 proceeds essentially in three steps. After having isometrically embedded M into some Euclidean space R k , for some k ∈ N (Step 1), we prove an analogous approximation result (Proposition 3.8) for unconstrained maps, i.e., for maps with values into R k (Step 2). Then, we obtain Theorem 3.1 from Proposition 3.8 by means of a suitable retraction onto M (Step 3).
As mentioned in Section 2.1, Step 1 can always accomplished by exploiting Nash's isometric embedding theorem. (However, since M is compact the choice of the embedding is essentially irrelevant, as it can only imply, at worst, enlarging the constant in (3.3).) Step 2 is heavily based on arguments in [16] and worked out in Section 3.1 below. We complete the proof of Theorem 3.1 in Section 3.2.

Remark 3.2.
The factor 1 + ρ 2 at right hand side in (3.3) is due to the use of the embedding theorem for variable-exponent Lebesgue spaces (Proposition 2.3) in the proof of Proposition 3.3 below. For constant exponents, this is not needed and one can instead stick with the clever, optimal argument in [16] and obtain (3.3) without the extra-factor 1 + ρ 2 , c.f. (3.16), (3.17) below. However, the dependence on ρ at right hand side of (3.3) is harmless for our purposes in this paper, as we apply Theorem 3.1 only to maps defined on a fixed, common ball (c.f. the blow-up analysis in the proof of Theorem 4.3).

The unconstrained case
For the unconstrained case, we follow very closely the argument of [16, Section 2 and Section 3], developed in the setting of constant exponents and functions of class SBD p . As mentioned at the beginning of the section, the main result here is Proposition 3.8. The key tool in the proof of Proposition 3.8 is Proposition 3.3 below. The statement of Proposition 3.3 is similar to that of the corresponding result in [16], i.e., [16, Theorem 2.1], up to the complications arising because of the variable exponent. In particular, we shall need the following assumption (H): This assumption is always satisfied in this work, because we deal with maps with values into a compact Riemannian manifold M and moreover, as mentioned in the Introduction and explained in more details in Section 3.2 below, to approximate them with maps with values into M (which is only assumed to be connected, in general) we need to assume p + < 2.
Assumption (H) ensures that every function of class W 1,1 with L p(·) -integrable gradient is also of class W 1,p(·) and, in turn, that functions belonging to SBV p(·) with no jump are also of class W 1,p(·) (see the discussion in Section 2.4 and the end of Step 5 below for more details on this point). Finally, we notice that also in the constant-exponent case the regime p < n is the most interesting one (and that the condition p < p * is trivially satisfied in such case). Proposition 3.3. Let p : R 2 → (1, +∞) be a bounded, log-Hölder-continuous, variable exponent satisfying p − > 1 and let k ∈ N. There exist a universal constant η and a constant c k , depending only on k, such that for any r > 0, if J ∈ B(B 2r ) is any Borel set in B 2r that satisfies

4)
then there exists R ∈ (r, 2r) for which the following holds. Let u ∈ SBV p(·) B 2r , R k be any function such that and suppose, in addition, that assumption (H) holds. Then, there exists φ(u) ∈ SBV p(·) B 2r , R k ∩ W 1,p(·) B R , R k such that the following properties are satisfied.
where the implicit constant at right hand side depends only on the log-Hölder constant of p.
where the implicit constant at right hand side depends only on k.
Proof. The strategy of the proof is borrowed from [16, Theorem 2.1], which deals with maps of class SBD p (B 2r ), for 1 ≤ p < +∞ a constant exponent. In rough words, the idea is to choose a "good radius" R ∈ (r, 2r) for a ball B R , concentric with B 2r , and to construct a grid B R by triangles, refining towards the boundary of B R , whose vertices are chosen in such a way to satisfy essentially three properties: (i) they are Lebesgue points of the function u to approximate; (ii) the one-dimensional restriction of u to each edge of the grid is of class W 1,1 , and (iii) the value of u at the vertices is close to the local average. (See, more precisely, properties (P 1 )-(P 5 ) below.) A crucial point is that, in 2D, such a grid can be constructed with universally bounded geometry under the mere assumption (3.4).
Once the grid has been built, one can then define a piecewise affine map φ(u) in B R by prescribing that it agrees with u at the vertices of each triangle of the grid. Outside B R , one sets φ(u) := u, and it turns out that the map φ(u) so defined satisfies all the properties in the statement.
Here, proceeding in a sequence of steps, we modify the approach of [16] to handle the complications arising because of the variable exponent, at the same time benefiting from some simplifications occurring because we deal with maps of class SBV. The main idea is to exploit the constancy of ∇φ(u) in each triangle of the grid to make a sort of local "L ∞ − L p − -interpolation", to estimate locally the p(·)-modular of ∇φ(u), c.f. Step 4. Then, we obtain global estimates summing over all triangles of the grid. The price to pay is the slight loss of exponent at right hand side in (3.6). Nevertheless, the estimate (3.6) will be enough to our purposes, i.e., for the blow-up analysis in Theorem 4.3.
Step 1 (Construction of the grid). Following the argument in [16, Theorem 2.1], we can pick R ∈ (r, 2r) in such a way that the following holds: and (C.f. [16, (2.2) and (2.3)].) Once R has been chosen as in the above, we construct a triangular grid T for B R , refining towards the boundary of B R , in a completely analogous way to as in first part of the proof of [16, Theorem 2.1]. We recall the main properties of T and the main steps of its construction, referring the reader to [16, pp. 1340-1344] for full details.
The construction of T starts in a completely geometrical way, by considering the circles of radii R h := R − δ h , with h ∈ N and δ h := R2 −h , and, for each such circle, the 2 h distinguished points x ′ h,j given by As in [16, p. 1341], we observe that any pair x ′ , y ′ of neighbouring points in (the notion of "neighbouring points" in V ′ coincides with the intuitive one and it is formalised where c 1 , c 2 > 0 are universal constants. Connecting all such vertices, we obtain a grid T ′ with universally bounded geometry. Now, we fix any function u ∈ SBV p(·) B 2r , R k such that assumption (H) is satisfied (so, if p does not satisfy (2.11), we assume in addition that u is bounded) and such that H 1 (J \ J u ) = 0. We are going to adapt the grid T ′ to the function u, obtaining a good grid T on which constructing a piecewise linear approximation of u.
To the claimed purpose, we follow the argument in [16] and we start by taking any triangle T ′ of the grid T ′ . We denote x ′ , y ′ , z ′ be the vertices of T ′ , and by s x ′ ,y ′ , s x ′ ,z ′ , s y ′ ,z ′ the segments connecting them. We define 11) and, in the same way, the sets Q x ′ ,z ′ , Q y ′ ,z ′ . Next, we define and we consider, for each pair of indexes (h, j), the ball B(x ′ h,j , αδ h ), of radius αδ h and center x ′ h,j . Let x ′ , y ′ any two neighbouring points in V ′ . The sophisticated iterative procedure in [16, pp. 1341-1344] shows that one can determine two universal constants η > 0 (bounding from above the ratio H 1 (J) /(2r)) and c > 0 so that there is a "good" set G ⊂ B (x ′ , αδ h ) × B (y ′ , αδ h ) with positive measure and whose points (x, y) satisfy the following properties: where u x ′ ,y ′ denotes the average of u in Q x ′ ,y ′ .
(P 5 ) x and y are Lebesgue point of u, In the above, u ν z denotes the 1D-section of u along the direction ν determined by x, y, given by These properties are exactly the analogues of properties (P 1 )-(P 5 ) in [16, Theorem 2.1] and they are proven with exactly the same reasoning as in [16], just replacing the symmetric gradient with the full gradient of u and the rigid motions with averages.
Arguing word-by-word as in [16, p. 1344], we can extract out of G a set of points V which are good vertices for a new triangulation T, which is adapted to u in the following sense: connecting all neighbouring points in V by edges (agreeing, as in [16], that two points in V are neighbours if and only if they come from points that were neighbours in the previous sense), we obtain a grid T by triangles T whose vertices satisfy (P 1 )-(P 5 ). (In particular, the edges of T do not intersect the jump set of u.) In addition, any pair of neighbouring vertices in V satisfy (3.10), and hence the angles of the triangles T are uniformly bounded away from 0 and π.
Moreover, for later purposes, we observe that the triangles T ′ of T ′ and the triangles T of T are in a one-to-one correspondence. For any given T ∈ T, we denote C T the following convex envelope: where the union runs over the vertices x ′ of the triangle T ′ in T ′ which correspond to T . Clearly, O x ′ ,y ′ ⊂ C T for any two vertices x ′ , y ′ of T ′ and T ⊂ C T . Moreover, following [16], we observe that there is a universal constant κ such that any x ∈ B R belongs to at most κ of the convex envelopes C T . In addition, we remark that, by construction, there exists two universal constants λ 1 , λ 2 > 0 such that (3.13) Let us set λ := max{λ 1 , λ 2 }.
Step 2 (Definition and main properties of the approximating map). We define, as in [16], a function φ(u) by letting φ(u) = u in B 2r \ B R while we take φ(u) to be the piecewise affine function determined by V on B R . More precisely, in each triangle T ∈ T, we define φ(u) as the (uniquely determined) affine function obtained by setting φ(u)(x) = u(x), for each vertex x of T . In particular, on the edges of T , φ(u) is the linear interpolation between the values of u at the vertices and ∇φ(u) is a constant (k × 2)-matrix in each T ∈ T. As a consequence, given any vertices x, y of T , we have With the aid of the fact that u ν z ∈ W 1,1 (s x,y ) and of (3.12), we can compute the elements of the matrix ∇φ(u) and its L ∞ -norm in complete analogy with [16], obtaining the following counterparts of [16, (2.12) and (2.13)]. For each T ∈ T, every pair x, y of vertices of T , and every j ∈ {1, . . . , k}, by the fundamental theorem of calculus on s x,y and (3.14) we have hence, by (3.12) and (3.13), Thus, letting x, y vary in the set of vertices of T and j run in {1, . . . , k}, for each T ∈ T. From (3.16) and Jensen's inequality, we obtain for any q ∈ [1, p − ] and any T ∈ T. Recalling that the sets C T overlap at most κ-times, and that κ is a universal constant, by taking the sum of (3.17) over T ∈ T it follows from (2.9) that φ(u) belongs to In addition, by construction of the grid, the BV trace of u on ∂B R and the Sobolev trace of φ(u) on ∂B R agree H 1 -a.e., hence no additional jump is created by the above process (see [16, p. 1346] for details). Consequently, φ(u) has less jump than u in B 2r .
Step 3 (Proof of (i), (iii), (iv), (v)). Item (v) is obvious. Item (i) follows immediately, because of (3.8) and since H 1 (J \ J u ) = 0. The first assertion in (iv) follows by definition of φ(u) and the argument in [16, p. 1346]. The second assertion in (iv) follows because, by construction, J φ(u) = J u on B 2r \ B R and, as observed in the previous step, u and φ(u) agree H 1 -a.e. on ∂B R .
To prove (iii), we argue similarly to as in [16]. Let T ′ be any triangle of the original, geometrical grid T ′ and let x ′ , y ′ , z ′ be its vertices, connected by the segments s x ′ ,y ′ , s x ′ ,z ′ , s y ′ ,z ′ . Let Q x ′ ,y ′ be defined as in (3.11). By Proposition B.1 and (3.10), we have where u x ′ ,y ′ denotes the average of u in Q x ′ ,y ′ and c k := c(k)c 2 , where c(k) ≡ C(2, k) is the constant, depending only on k, provided by Proposition B.1 (for n = 2) and c 2 is the constant in (3.10). Moreover, by (P 4 ), it follows that where the implicit constant at right hand side is universal.
Next, for every T ∈ T, let T ′ ∈ T ′ be the corresponding triangle in the original grid, with vertices x ′ , y ′ , z ′ . By triangle inequality we havê (3.20) About the first term above, we notice that, again by triangle inequality, there holdŝ where u T denotes the average of u in T . For the first term on the right in (3.21), by Proposition B.1 and (3.10), we get where, again, c k = c(k)c 2 . To estimate the second term in (3.21), we first bound the constant u T − u x ′ ,y ′ using (3.18) and (3.22) as follows: Thus,ˆT Thus, from (3.13), where c k := c k λ is a constant depending only on k. Combining (3.21), (3.22), and (3.23), we obtainˆT where c k depends only on k.
Concerning the second term in (3.20), we proceed in two steps. First, we observe that the function u x ′ ,y ′ − φ(u) is an affine function, and hence its modulus achieve its maximum at one of the vertices of T . Denoting ξ such a vertex, we havê If instead ξ = z, then we write The second term above is again estimated by (3.19), which yields where the constant c is universal. Next, we estimate the constant u x ′ ,y ′ − u x ′ ,z ′ as follows: where in the last line we used again Proposition B.1 and the implicit constant depends only on k. Therefore, using again (3.13) where the implicit constant at right hand side depends only on k. Therefore, taking the sum of the above inequalities as T ranges over T (recalling once again that the sets C T have finite overlap), we obtain (3.7).
Step 4 (L ∞ -L p − interpolation). We notice that if Ω ⊂ R n is any measurable set, f ∈ (L p − ∩ L ∞ ) Ω, R k is arbitrary, and p(·) is any bounded, variable exponent, with 1 ≤ p − ≤ p(x) ≤ p + < ∞, then for a.e. x ∈ Ω there holds Step 5 (Conclusion). For each T ∈ T, we apply (3.30) with Ω = T and f = ∇φ(u) (which is constant in T , being φ(u) affine in T ). By (3.16), (3.30) and Hölder's inequality, we obtain where C T is defined as in Step 3. On the other hand, using again (3.16), where the implicit constant at right hand side is universal. By (3.30) and (3.31), where the last inequality follows from the log-Hölder condition (2.5) (applied to the smallest ball B T containing C T ) and ℓ is precisely the constant in (2.5). Thus, by (3.32) and Proposition 2.3,ˆT up to a universal constant. The inequality (3.33) clearly implies (3.6) at the local level. Taking the sum of the inequalities (3.33) as T varies in T (recalling again that the sets C T have finite overlap), we obtain (3.6), completing the proof of (ii). By assumption (H) and either (2.10) or Lemma 2.10, it follows that,ˆB R |φ(u)| p(x) dx < +∞, and therefore, by (3.6), φ(u) ∈ SBV p(·) B 2r , R k ∩ W 1,p(·) B R , R k . This concludes the proof.

Remark 3.4. On the account of the inequalities (2.3), inequality (3.6) readily implieŝ
where the implicit constant depends only on the log-Hölder constant of p. Consequently, since φ(u) = u in B 2r \ B R , the same inequality holds with 2r in place of R.
Remark 3.5. The inequalities (3.6) and (3.34) are far from being sharp, both because of the loss of exponent in Step 4 above and because we did not try to optimise with respect to the constants at right hand side. Nonetheless, they are enough for our purposes in this paper. Remark 3.6. As in [16], Proposition 3.3 and, consequently, Proposition 3.8 and Theorem 3.1 below, is valid only in 2D. This is due to the fact only in 2D a condition like (3.4) is strong enough to ensure that the jump set can be avoided by a grid made by simplexes with universally bounded edges and angles. To obtain a similar outcome in the higher dimensional setting, it seems to be really necessary to exclude a "small" set from the domain, along the lines of [7]. See Appendix C for more details on these points.
Next, along the lines of the argument in [16], we use Proposition 3.3 and the SBV p(·) compactness theorem (Corollary A.3) to prove Proposition 3.8 below (which is the counterpart of [16, Proposition 3.2]). Before stating the theorem, we recall from [16] the following useful lemma.
We recall the a set E ⊂ R n is H d -rectifiable if and only if it is countably H d -rectifiable and H d (E) < +∞, see [4, Definition 2.57]. For the reader's convenience, we provide a detailed proof of Lemma 3.7, because this gives us the occasion to fix some small drawbacks of that in [16] and to provide some additional details.
Proof. Since J is a Borel set with finite H 1 -measure, the restricted measure H 1 J is a Radon measure in R 2 . Fix any x ∈ J ∩ B 2sρ . Then, for all λ ∈ ((1 − s)ρ, 2(1 − s)ρ) except at most countably many we have H 1 J ∩ ∂B λx/2 k (x) = 0 for all k ∈ N (see, e. g., [4, Example 1.6.3]), i.e., for which (3.35) holds. Choose any such λ x ∈ ((1 − s)ρ, 2(1 − s)ρ) and define Since J is a Borel set in R 2 and it is H 1 -rectifiable, the H 1 -density of J is 1 at H 1 -almost every x ∈ J (for instance, by Besicovitch-Marstrand-Mattila theorem, see, e. g., [4, Theorem 2.6.3]). Clearly, J ∩ B 2sρ is H 1 -rectifiable as well. Therefore, the set Thus, we would have Since η < 1, we would obtain that, for any x ∈ E, the H 1 -density of J ∩ B sρ at x is less than 1, a contradiction. Thus, for H 1 -a.e. x ∈ J ∩ B 2sρ , the set I x has a least upper bound, which is actually a maximum, because the only possible accumulation point of I x is 0. If the max in the definition of r x is attained for k ≥ 2, then (3.36) holds by definition. If instead r x is attained for k = 1, then 2r x = λ x and (3.36) holds as well, otherwise a contradiction. Thus, (3.36) is verified. This concludes the proof. There exist universal constants ξ, η > 0 such that the following holds. Provided that Assumption (H) is satisfied, then for any s ∈ (0, 1) and any u ∈ SBV p(·) B ρ , R k satisfying there are a countable family F = {B} of closed balls, overlapping at most ξ times, of radius r B < (1− s)ρ and centre x B ∈ B sρ , and a function w ∈ SBV p(·) B ρ , R k such that (iv) w ∈ W 1,p(·) B sρ ; R k and H 1 (J w \ J u ) = 0.
(v) For each B ∈ B, one has w ∈ W 1,p(·) B, R k , witĥ for any q ∈ [1, p − ], where the implicit constant at right hand side is universal. Moreover, where the implicit constant in the right hand side depends only on the log-Hölder constant of p.
Proof. With Proposition 3.3 at hand, the proof is completely analogous to that of [16, Proposition 3.2]. We sketch the key steps to point out the main differences, addressing the reader to [16] for the missing details.
Let s ∈ (0, 1) be fixed and set J = J u . Then, J is a Borel set in B ρ , H 1 -rectifiable, and such that H 1 (J ∩ B ρ ) < η(1 − s)ρ/2. According to Lemma 3.7, the set E of points x ∈ J u ∩ B sρ at which at least one between (3.35) Moreover, since all the radii r B of the balls B in F are smaller than ρ, we get F r 2 ) follows, too. By (3.35), there holds H 1 (J u ∩ ∪ F ∂B) = 0, and this completes the proof of (ii).
On each ball B i 1 of the first family F ′ 1 , we consider the function φ i 1 (u) given by Proposition 3.3. For each h ∈ N, we define Properties (ii) and (iv) in Proposition 3.3 imply that and In addition, by (ii) in Proposition 3.3, it also follows that, for any h ∈ N, up to a constant depending only on the log-Hölder constant of p(·). Moreover, for any h ∈ N there holds for a universal constant c > 0. Furthermore, by (iii) of Proposition 3.3, and, for any j ≥ h ≥ 1, where c k > 0 depends only on k. Thus, as in [16,Proposition 3.2], we see that the sequence and φ(u) := i∈N φ i 1 (u)χ B i 1 . By (ii), (iii) in Proposition 3.3, it follows that φ(u) ∈ SBV p(·) B ρ , R k ∩ W 1,p(·) ∪ F ′ 1 B, R k . Therefore, by (3.41), (3.40), and Corollary A.3, we have w 1 ∈ SBV p(·) B ρ , R k . Exactly as in [16, end of p. 1349], since we conclude that and therefore, by (2.10) or Lemma 2.10 (depending on which of the two conditions in Assumption (H) is verified), it follows that w 1 ∈ W 1,p(·) ∪ i∈N B i 1 . In addition, by construction, w 1 = u L 2 -a.e. on B ρ \ ∪ F ′ 1 B and H 1 (J w 1 \ J u ) = 0. Iterating the previous construction for any integer l with 1 < l ≤ ξ, considering the l-th family F ′ l , the sequence otherwise, and its L 1 -limit

Remark 3.10. As an obvious consequence of items (i) and (iii) of Theorem 3.8 and of (2.3), it follows that
where the implicit constant at right hand side depends only on the log-Hölder constant of p(·).

The constrained case
In this section, we prove Theorem 3. Lemma 3.11. If M is a compact m-dimensional smooth submanifold of R k with π 0 (M) = π 1 (M) = · · · = π j (M) = 0 then there exist a compact (k − j − 2)-dimensional smooth complex X in R k and a locally smooth retraction P : R k \ X → M so that for any y ∈ R k \ X and some constant C = C(M, k, X). Moreover, in a neighborhood of M, P is smooth of constant rank m.

Remark 3.12.
By construction, X is strictly away from M. Apparently, Lemma 3.11 was firstly proven by Hard and Lin [31, Lemma 6.1], with X a Lipschitz polyhedron and P locally Lipschitz retraction. Later, it was realised that the same argument, at price of very minor complications, allows to construct X as a smooth complex and P as a locally smooth retraction, c.f., e.g., [6,8]. We import almost verbatim a useful remark from [8].  Let Ω ⊂ R n be a bounded open set with Lipschitz boundary. Let j ≥ 0 be an integer, let k ∈ N and assume that M is a compact, j-connected, smooth Riemannian manifold without boundary, isometrically embedded into R k . Let p : Ω → (1, +∞) be a bounded, variable exponent, satisfying for every x ∈ Ω. Then, for any w ∈ L ∞ ∩ W 1,p(·) Ω, R k so that w| ∂Ω takes values in M, there exists w ∈ W 1,p(·) (Ω, M) such that w = w a.e. on {x ∈ Ω : w(x) ∈ M} and w = w on ∂Ω (in the sense of traces). Moreover,ˆΩ

46)
where C > 0 is a constant independent of w and w and that depends only on the quantities listed in Remark 3.17.
The proof of Lemma 3.16 follows very closely arguments and computations in that of [31, Theorem 6.2], taking advantage of several observations in [8]. For the reader's convenience, we provide full details. During the proof, we will keep track of the precise dependencies of the constant C in (3.46), gathering them in Remark 3.17 below.
Proof. Let P : R k \X → M be given by Lemma 3.11. For a ∈ R k , define X a := {y + a : y ∈ X} and P a : R k \ X a → M by P a (y) : y ∈ R k \ X a → P(y − a). By Lemma 3.11, the map P a is well-defined and locally smooth and, by Remark 3.14, we may find σ > 0 so small that the restricted map P a | M : M → M is a diffeomorphism for all a ∈ B k σ := y ∈ R k : |y| < σ . Up to further reducing σ below a threshold valueσ (depending only on P and M), if necessary, the inverse function theorem implies that λ := sup a∈Bσ Lip (P a | M ) −1 is a finite number depending only on P and M.
Now, we notice that the set By Fubini's theorem, it follows that H n (N ∩ (Ω × {a})) = 0 for a.e. a ∈ R k , so P a • w is well-defined for a.e. a ∈ R k and it is a measurable function. Moreover, by the chain rule, for a.e. (x, y) ∈ Ω × R k we have Applying Lemma 3.15 with v = w, f = |∇P| p(x) , g = |∇w| p(x) , Λ = w L ∞ (Ω) and σ > 0 as above, we obtain where in the last line we used the obvious inequality |∇P(y)| p(x) ≤ 1 + |∇P(y)| p + (which holds for almost every (x, y) ∈ Ω × R k ) and Remark 3.17. Thus, the constant C ′ depends only on p + , M and P (through j, k, and´Bσ |∇P| p + dy, where the latter is finite by (3.44), since p + < j + 2). Therefore, by Chebyshev inequality, we may choose a ∈ B k σ so that where, by Remark 3.17, C > 0 is a constant that depends only on p + , p − , k, M, P, where P is the retraction provided by Lemma 3.11. Upon setting we see that w ∈ W 1,p(·) (B sρ , M) ∩ SBV p(·) (B ρ , M) and that H 1 (J w \ J u ) = 0 (indeed, in view of (3.48), H 1 (J w \ J v ) = 0). Moreover, w| ∂Ω = w s | ∂Ω . Now, we claim that the function w and the family F are a function and a family of balls satisfying all the properties required by the statement. Indeed, item (i) is obvious (as it already follows from Proposition 3.8). Item (ii) holds because u = v a. e. on B\∪ F B, so that in particular v s = u on B sρ \∪ F B, and w s coincides with v s when the latter takes values in M (in particular, on the set {u = v} ∩ B sρ ). Item (iii) is an immediate consequence of the above discussion. Finally, item (iv) follows from (3.49), (3.39), and the definition of w.

Remark 3.19.
For simplicity, we stated Theorem 3.1 assuming that M is merely connected. However, a quick inspection of the proof shows that it is valid, with identical proof, if we assume M is j-connected and we replace the constraint p + < 2 with p + < j + 2.

Regularity of local minimisers
In this section, we prove Theorem B. We will follow ideas that originated in [25] in the scalar case and that were later adapted to the context of sphere valued maps in [11]. Differently from [25,11], we will not use the Sobolev-Poincaré inequality for SBV functions and we will not make use of medians or truncations. Instead, we will follow the approach of [17] and employ the Sobolev approximation from Section 3. As explained in the Introduction, the key point lies in proving that the jump set of local minimisers is essentially closed.
Let Ω ⊂ R 2 be a bounded open set and let k ∈ N. In this section, we shall need the following variants of the integral functional F in (1): for A ⊂ R 2 a Borel set, we consider  For every open set A ⊂ R 2 , every t > 0 and every u ∈ SBV loc Ω, R k such that |u| = t a.e. in Ω, we define If Φ(u, c, A, t) < +∞, we define the deviation from minimality of u as Equivalently, Dev(u, c, A, t) can be characterised as the smallest number κ ∈ [0, +∞] so that for all v ∈ SBV loc Ω, R k satisfying {v = u} ⊂⊂ A and such that |v| = t a.e. in Ω.
For convenience, when c = 1, we set Similarly to as in [4,Section 7], we establish the following definition.

Definition 4.2.
A function u ∈ SBV loc Ω, S k−1 is a quasi-minimiser (among maps with values into S k−1 ) of the functional F (·, Ω) in Ω if there exists a constant κ ′ ≥ 0 such that for all x ∈ Ω and all balls B ρ (x) ⊂ Ω, By a standard comparison argument, we get an immediate upper bound on the local energy of quasi-minimisers, that will be used in the proof of Theorem 4.2 below.
Proof. The proof is almost identical to that of [4,Lemma 7.19]; the only difference is that we are compelled to consider competitors with values into S k−1 . (See also [11,Lemma 3.2] and the remark thereafter.) Let u ∈ SBV loc (Ω, S k−1 ) be a quasi-minimiser of F (·, Ω) in Ω, take any ρ ′ > 0 with ρ ′ < ρ and put v : and, by quasi-minimality of u, The conclusion follows by letting ρ ′ → ρ.
To obtain that J u is essentially closed, we again argue along the lines of [25,11,17,34]. The heart of matter is proving, in any set Ω δ ⊂ Ω defined as in (2.2), inequality (4.5) below.

Theorem 4.2 (Density lower bounds).
Let p : R 2 → (1, +∞) be a variable exponent satisfying (p ′ 1 ) and (p 2 ). Let δ > 0 and Ω δ be defined by (2.2). There exist θ δ and ρ δ depending only on p − , p + , and δ with the property that, if u ∈ SBV p(·) Ω, S k−1 is a quasi-minimiser of F in Ω, then for all balls B ρ (x) ⊂ Ω δ with centre x ∈ J u and radius ρ < ρ ′ δ := ρ δ κ ′ , where κ ′ is the constant in (4.4). Moreover, Once again, the idea of the proof of Theorem 4.2 goes back to De Giorgi, Carriero, and Leaci [25]. It is based on proving a power-decay property of the energy in small balls with respect to the radius of the balls which, after a classical iteration argument, yields the conclusion. Given such a decay property, that in our case is provided by Theorem 4.3 below, the proof is purely "algebraic" and it does not depend in any way on the target manifold of the considered maps. However, some modifications are needed with respect to the argument in [25,4] and also with respect to the recent adaptation to the variable-exponent setting provided by [34,Theorem 4.7] because we need to avoid recurring to medians and truncations. The key technical tool that allows us to do so is provided by Proposition D.1 in Appendix D, which gives us a sufficient condition to conclude that a given point does not belong to the jump set of a map u and that is proved using only the Sobolev approximation results from Section 3. Once Proposition D.1 is obtained, the proof of Theorem 4.2 follows by the classical argument in [25], exactly as in [4,34].
The bulk of this section is instead devoted to proving Theorem 4.3, a task which requires to modify in a nontrivial way both the arguments in [34] and those in [11].

Theorem 4.3 (Decay lemma).
Let p : R 2 → (1, +∞) be a variable exponent satisfying (p ′ 1 ) and (p 2 ), let δ > 0, and let Ω δ be defined as in (2.2). There exists a constant C δ = C(p − , p + , δ) > 0 with the following property. For every τ ∈ (0, 1) there exist ε = ε(τ, δ) and θ = θ(τ, δ) in (0, 1) such that, if u ∈ SBV(Ω, S k−1 ) satisfies, for all x ∈ Ω δ and all σ < ε 2 such that B σ (x) ⊂⊂ Ω δ , the conditions The idea of the proof of Theorem 4.3 is classical and rooted in an argument by contradiction. However, the combined non-standard growth of the energy functional and the fact that we deal with maps with values into spheres make it somewhat trickier than usual. To handle the first complication, i.e., the non-standard growth of the functional, we borrow some ideas from the recent work [34], where the functional F is considered on scalar-valued functions of class SBV p(·) . Note that our argument requires, as in [34], the variable exponent to satisfy the strong log-Hölder condition, c.f. (4.15) and Remark 4.8 below.
To take the sphere-valued constraint into account, we reason as in [11,Section 3] but, instead of using medians, truncations, and the Sobolev-Poincaré inequality for SBV functions, we exploit the Sobolev approximation results from Section 3. Nonetheless, for the proof of Theorem 4.3 we shall need several auxiliary results in the spirit of [11]. The proof of Lemma 4.4 is straightforward and, as in [11], left to the reader.
for some constant d ≥ 0.
Then, there exists a hyperplane Π ⊂ R k such that w ∞ (x) ∈ Π for L 2 -almost every x ∈ B.
Proof. The proof is along the lines of that of the corresponding assertion in the proof of [11,Theorem 3.6]. Up to rotations in the target space, we may assume that for any h ∈ N there holds where e k denotes the last vector of the canonical basis of R k . Following [11], we observe that the trivial identity w h − λ Hence, by (4.8) (which implies lim and assumption (ii) we obtain 0+2w 0 ·e k = 2d hence w 0 · e k = 0 a.e. in B. The conclusion follows.
For later use, we record here the following trivial linear algebra lemma. Proof. By assumption, the hyperplane Π (1/2) is a vector subspace, of dimension r ≤ k − 1, contained in all the hyperplanes Π (s) . Let {a 1 , . . . , a r } be r affinely independent vectors in Π (1/2) . If r = k − 1 or r coincide with the maximal dimension of the hyperplanes Π (s) , we are done. So, let us assume r < k − 1 and that there is s > 1/2 such that Π (1/2) is a proper subspace of Σ (s) . Let r (s) ≤ k − 1 be the dimension of Π (s) . Then, in Σ (s) we can find r (s) − r vectors a r+1 , . . . , a r (s) which are affinely independent both each other and from a 1 , . . . , a r . Again, if r (s) = k − 1, we are done, otherwise we repeat the process. In at most k − r − 1 steps, we get the conclusion.
Finally, we recall the following L ∞ -L 1 estimate for p-harmonic functions, see [37,29]. We report here a statement which follows as a special case of a more general result proved for differential forms.  We are now ready to prove Theorem 4.3. As already mentioned, the argument combines ingredients from [11], [17], and [34].
We now proceed step-by-step, following the argument in [34].
Step 1 (Scaling, Part 1). For all h ∈ N, we define the translated maps and exponents where y ∈ B 1 . We notice that for any h ∈ N and any y ∈ B 1 Moreover, each rescaled variable exponent p h (·) is still strongly log-Hölder continuous and this implies that the sequence {p h } converges to p 0 uniformly in B 1 as h → +∞. Indeed, upon setting which tends to zero as h → +∞ thanks to (2.6). Concerning the translated maps u h , we observe that´B where the second equality follows from (4.12) and the obvious fact that Next, we set We define the rescaled functions and we observe that, for all h ∈ N, . From (4.11) and the definition of γ h , there holds Step 2 (Scaling, Part 2). For any h ∈ N, we define From the definitions of p h and v h , Equations (4.12), (4.13), (4.14) become, respectively, where Dev h denotes the deviation from miniminality related to F h . By (4.17), there holds and, since γ h σ h ≤ 1, we also havê Step 3 (Approximation and lower bound). For any s ∈ [1/2, 1), thanks to (4.17), we can use Theorem 3.1 to associate with any u h (for any h sufficiently large), a function z (s) h belonging to W 1,p h (·) B s , S k−1 ∩ SBV p h (·) B 1 , S k−1 and a family of balls F (s) h satisfying, in particular, where the implicit constant at right hand side depends only the log-Hölder constant of p(·) and, by (4.17) and (3.2), as h → +∞. Moreover, Next, for any s ∈ [1/2, 1) and any sufficiently large h ∈ N as in the above, we denote We observe that z (s) for any h and any s ∈ [1/2, 1) and that each map z (s) h coincides with the Sobolev approximation of v h provided by Theorem 3.1. In particular, and, by (4.23) and (3.2), for any s ∈ [1/2, 1) we have as h → +∞. Moreover, for any s ∈ [1/2, 1), there holds We claim that, for any s ∈ [1/2, 1), for some d ≥ 0. Indeed, the second equality entails the first and we have, for any s ∈ [1/2, 1), where we used that z (s) h = 1 a.e., Jensens's inequality, the classical Sobolev-Poincaré inequality, and Proposition 2.3. By (4.30), it follows, in particular, that for any q ∈ [1, (p − ) * ], up to a constant independent of s. Furthermore, we observe that Indeed, by (4.28) and (4.29),
In particular, we have ∇z As (4.32) holds for any s ∈ [1/2, 1), we can let s ↑ 1 in the above inequalities and find botĥ In addition, Lemma 4.5 applied to the sequence z (s) yields that each map z (s) takes value in a (proper) hyperplane Π (s) , that we may assume to be contained in the hyperplane {x k = 0}. Moreover, by Lemma 4.6, we can find a maximal hyperplane Π containing all the Π (s) . Finally, since p − < 2 and p + < (p − ) * , we have z ∈ W 1,p 0 (B 1 , Π), by (4.32), Poincaré's inequality and the fact that p 0 ≤ p + .
Step 4 (Convergence of the energy and p 0 -minimality of z). We now improve the lower bound in the previous step by showing that z is a local minimiser of the p 0 -energy with respect to compactly supported perturbations.
We start by noticing that, since the function s → F h (v h , γ h , B s ) is increasing and uniformly bounded for s ∈ (0, 1], by Helly's selection theorem and possibly passing to a (not relabelled) subsequence, we may assume that the limit exists for all s ∈ (0, 1], where the function s → α(s) is non-decreasing. (Hence, continuous for all but at most countably many s ∈ (0, 1].) Let v ∈ W 1,p 0 (B 1 , Π) be such that {v = z} ⊂⊂ B 1 . Let {v ε } ⊂ W 1,∞ B 1 , R k be any sequence of Lipschitz functions strongly converging to v in W 1,p 0 (B 1 ). Take ρ ′ , ρ ∈ (0, 1), with ρ ′ < ρ and ρ a continuity point of α(·). For any h large enough, by (4.21) we have where η is the universal constant in Theorem 3.1, and therefore the approximation z (ρ) of v h , constructed as in the previous step, is well-defined. Let ζ ∈ C ∞ c (B ρ , [0, 1]) be a cut-off function with ζ ≡ 1 in B ρ ′ and |∇ζ| ≤ 2 ρ−ρ ′ . We may assume that {v = z} ⊂⊂ B ρ and define (for h large enough) h in the sense of (Sobolev) traces on ∂B ρ . By writing we see that, for any small enough ε > 0 and any h ∈ N large enough, Indeed, v · e k = z · e k = 0 because v and z take values in Π, v ε (y) → v(y) at a.e. y ∈ B 1 as ε → 0 by strong convergence, and z + λ → 0 a.e. in B ρ as h → +∞, which follows as a consequence of the weak convergence of z (ρ) of the Sobolev-Poincaré inequality (4.31). This also implies that z ε) a. e. for h large enough, which yields the claim.
As a consequence of the above discussion, for any ε > 0 small enough and any h ∈ N large enough, we can define and, in turn, (4.36) For any fixed ε > 0, the sequence |∇v ε | p h (·) is equibounded in B 1 (because v ε is Lipschitz and {p h } converges uniformly), and therefore Consequently, by the strong convergence v ε → v in W 1,p 0 (B 1 ),  On the other hand, by the very definition of deviation from minimality, there holds for all h ∈ N. Thus, passing to the limit superior in (4.40) and using (4.19) (recalling that, by Lemma 4.4, the function s → Dev h (·, ·, ·, B s , ·) is increasing, for any h ∈ N), we obtain where the constant c is universal. Therefore, by (4.36), (4.37), (4.41), and the last two estimates, and letting ρ ′ → ρ (recalling that ρ is a continuity point of α(·)), As this holds for any ε > 0 (small enough), we can let ε → 0 and thanks to (4.38) we obtain Since we can take, in particular, v = z, we obtain Bρ |∇z| p 0 dy = α(ρ) (4.43) for all but countably many ρ ∈ (0, 1). In addition, since the left hand side of (4.43) is a continuous function of ρ ∈ (0, 1], we have that α(·) is actually a continuous function of ρ ∈ (0, 1]. Thus, the above argument holds for all ρ ∈ (0, 1) and we conclude that z minimises the p 0 -energy locally in B 1 with respect to compactly supported perturbations.

Remark 4.8.
We remark that, in the proof of Theorem 4.3, we used the strongly log-Hölder condition only to ensure the uniform convergence of the rescaled variable exponents p h (·) to a constant.
We are now in the position to prove Theorem 4.2.
Proof of Theorem 4.2. The proof is similar to that of [34,Theorem 4.7], which adapts to the variable exponent setting the classical argument from [25] (see also [4,Theorem 7.21]). We work out the main steps, addressing the reader to the aforementioned references for the missing details.
Step 3 (J u is locally essentially closed). We now prove that for any δ > 0. To this purpose, it is enough to prove that the difference between Ω δ ∩ J u \ J u and a H 1 -null set is itself a H 1 -null set. To do this, we argue as in the last part of the proof of [34,Theorem 4.7] and we consider the set Since |∇u| p(·) ∈ L 1 (Ω), it follows that H 1 (Σ δ ) = 0 (see, e.g., [28,Section 2.4
Step 4 (Conclusion). The essential closedness of J u , i.e., (4.6), now follows by an easy argument by contradiction. Assume H 1 Ω ∩ J u \ J u > 0 and let δ h ↓ 0 be any decreasing sequence (so that {Ω δ h } is an ascending sequence of sets such that Ω = ∪ ∞ h=1 Ω δ h ). Then, by the σ-subadditivity of H 1 and (4.48), We are now ready for the proof of Theorem B.

Remark 4.9.
In the particular case p is constant, we can apply standard elliptic regularity results [29] to obtain u ∈ C 1,β 0 loc Ω \ J u , S k−1 , for some β 0 ∈ (0, 1) depending only on k and p.
Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations
The authors have no competing interests to declare that are relevant to the content of this article.

A Closure and compactness theorems for SBV p(·)
The following results are certainly known to experts but we have not found any explicit proof in literature, hence we provide one here, for the reader's convenience.
Proof. The conclusion follows by combining the classical closure theorem for SBV (e.g., [4, Theorem 4.7]) with the lower semicontinuity result given by [22,Theorem 1.1] in the context of variable exponents. Although short, for clarity we divide the easy proof in two steps.
Remark A.2. Assume M is a smooth, compact Riemannian manifold without boundary, isometrically embedded in R k . If {u h } ⊂ SBV p(·) (Ω, M) and the assumptions of Theorem A.1 hold, then the limit function u provided by Theorem A.1 belongs to SBV p(·) (Ω, M). Indeed, the strong convergence u h → u in L 1 Ω, R k implies (up to extraction of a -not relabelledsubsequence) u h (x) → u(x) for a.e. x ∈ Ω, hence u(x) ∈ M for a.e. x ∈ Ω.
Corollary A.3 (Compactness theorem for SBV p(·) ). Let Ω ⊂ R n be a bounded open set and let p : Ω → (1, +∞) be a bounded, log-Hölder continuous variable exponent satisfying Let θ : [0, +∞) → [0, +∞] be an increasing lower semicontinuous function satisfying θ(t)/t → +∞ as t → 0. Finally, assume that {u h } ⊂ SBV p(·) Ω, R k is a uniformly bounded sequence in BV Ω, R k and that (A.1) holds. Then, we may find u ∈ SBV p(·) Ω, R k and extract a (not relabelled) subsequence such that u h → u weakly * in BV Ω, R k . Moreover, the Lebesgue part and the jump part of the derivative converge separately, i.e., ∇u h → ∇u and D j u h → D j u weakly * in Ω.
Proof. Again, we divide the simple proof in two steps.

B Poincaré's inequality for bounded variation functions in convex sets
Let Ω ⊂ R n be an open set and k ∈ N be an integer. We recall that every function in BV Ω, R k can be approximated by a sequence smooth functions. More precisely, the following characterisation holds (see, e. g., [ where the constant C(n, k) depends only on n and k.

C A counterexample in higher dimension
In this appendix, expanding on Remark 3.6, we provide an example which shows that the construction of the Sobolev approximation in Section 3 cannot work in higher dimensions under the mere assumption of smallness of the jump set.
Carefully looking at the proof of [16, Theorem 2.1], it is seen that the only point in which the assumption on the dimension is used is in the proof of property (P 2 ), and more precisely in the passage from [16, (2.8)] to [16, (2.9)], which relies on inequality (3.9) (i.e., on [16, (2.3)]). Again by inspection of the proof of (P 2 ), it is easily realised that, in order for the construction to work in higher dimensions, for any η ∈ (0, 1) and any given Borel set J ⊂ B n 2r with H n−1 (J) < η(2r) n−1 we must be able to find a radius R ∈ (r, 2r) satisfying both and where δ h := R2 −h and C n is a dimensional constant. Now, for every n ∈ N, given any Borel set J in B n 2r with H n−1 (J) < +∞, the measure H n−1 J is a Radon measure, hence equation (C.1) holds for all R ∈ (r, 2r) except at most countably (indeed, Radon measures can charge at most countably many boundaries of encapsulated sets; c.f., e.g., the discussion in [4, Example 1.63]). However, (C.2) may fail for n ≥ 3. We exhibit an example below for n = 3 (for ease of notation) that can be easily adapted to any dimension. Proposition C.1. For any ε ∈ (0, 1), any C > 0, and any r > 0, there exists a H 2 -rectifiable, starshaped, relatively closed and essentially closed Borel set J ⊂ B 3 2r with the following properties: (ii) There exists h 0 ∈ N so that for every R ∈ (r, 2r).
Proof. Fix arbitrarily ε ∈ (0, 1) and C > 0. By scaling, it suffices to consider the case 2r = 1. Let us set, for brevity, B := B 3 1 . Let {x j } be an enumeration of Q 3 ∩ ∂B. For any j ∈ N, let C (ε) j be the cone with apex the origin, axis the radius − − → Ox j , and opening angle ε2 −j 40π (the reason for this (ε) Then, u ∈ SBV p B 3 2r , R k for any p ≥ 1, J u = J, H 2 (J \ J u ) = 0, H 2 (J u ) < ε, J u is essentially closed and it does not satisfy (C.2).

D A criterion for being out of the jump set
In this appendix, we prove a sufficient criterion that allows for excluding that a point belongs to the jump set. An analogous, but slightly stronger, statement concerning the whole approximate discontinuity set S u is classical (see [25] and [4, Theorem 7.8]) but the classical proof uses medians and truncations, which we want to avoid. Here, we use only the tools provided by the Sobolev approximation results from Section 3.
Step 1. Suppose, for the sake of a contradiction, that x ∈ J u . Then, by definition (c.f., [4,Definition 3.67]), there exist a, b ∈ R k and ν 0 ∈ S 1 such that a = b and Moreover, the triple (a, b, ν 0 ) is uniquely determined, up to a permutation and a change of sign.
Observe that, thanks to the trivial pointwise inequality ∀ξ ∈ R k , |ξ| p − ≤ 1 + |ξ| p(x) , holding at each point x ∈ Ω, condition (D.1) implies Step 2. We are going to show that, given (D.2), we can find a sequence ρ j ↓ 0 as j → +∞ and m ∈ R k such that Now, take any sequence {ρ h } such that ρ h ↓ 0 as h → +∞. Then, for any h ∈ N, where σ h → 0 as h → +∞. For each h ∈ N so large that 2σ h < η, where η is the universal constant in Proposition 3.8, let s h := 1− 2σ h η . Then, let w (s h ) ρ h ∈ W 1,p − B s h , R k ∩SBV p − B 1 , R k be the approximation of u ρ h provided by Theorem 3.8. Recall that, since u (and hence each map ρ h is bounded as well and, indeed, w is uniformly bounded, the sequence m we can find a sequence {h j } j and m ∈ R k so that ρ h j → m as j → +∞.
Let us also set, for brevity's sake, for each j ∈ N.
Step 3. We claim that (D. By construction, the last term at right hand side above tends to 0 as j → +∞. As for (I) j , by items (i) and (iii) of Proposition 3.8 we havê as j → +∞. We now estimate (II j ), using that w j is a Sobolev function in B s j and item (v) of Proposition 3.8:ˆB as j → +∞, because s j → 1 as j → +∞ and thanks to item (v) of Proposition 3.8 and to (D.4). By the above estimates, (D.5) follows immediately.
Step 4 (Conclusion). By assumption, we should have |u| q dx < +∞ for some q > 1, which is exactly the hypothesis required in the original formulation of the criterion from [25] (in place of the boundedness of u). Up to the fact that we also need to assume (2.11) (in order to apply Proposition 3.8), the details are exactly as in [25] or, equivalently, [4, Theorem 7.8] and, therefore, omitted.

E Compactness of sequences of SBV p(·) -functions with vanishing jump set
Here, arguing along the lines of [17, Section 3], we use Theorem 3.1 and compactness results in variable exponent Sobolev spaces (see, e.g., [27,Chapter 8] Then, we can find a (not relabelled subsequence) and a function u ∈ W 1,p(·) (B ρ , M) such that u h → u in L p(·) (B ρ ) as h → +∞. Moreover, Proof. The proof is along the lines of [17,Proposition 3.2]. We reproduce the main steps, to point out the relevant changes.
For any s ∈ [1/2, 1) and any h ∈ N so large that H 1 (u h ) < η(1 − s), where η is the universal constant provided in Theorem 3.1, we let w  is bounded in W 1,p(·) B sρ , R k for any s ∈ [1/2, 1) and we can find a subsequence (depending on s and not relabelled) such that it converges to some w (s) weakly in W 1,p(·) B sρ , R k , strongly in L p(·) B sρ , R k (by [27,Corollary 8.3.2]), and pointwise a.e. on B sρ (because, in particular, we have weak convergence in W 1,p − B sρ , R k and hence strong convergence in L p − B sρ , R k by the classical Rellich-Kondrachov theorem).
By item (i) in Theorem 3.1, for any s ∈ [1/2, 1) and, correspondingly, any h sufficiently large, there holds where ξ is the same universal constant as in Theorem 3.1. By (E.1) and (E.5), it follows that w (s) = w (t) L 2 -a.e. on B sρ if 1/2 ≤ s ≤ t < 1. Thus, we can define a limit function u on B ρ by letting u = w (s) in B sρ for all s ∈ [1/2, 1). By definition, it is clear the u ∈ W 1,p(·) loc B ρ , R k . Moreover, for any s ∈ [1/2, 1), Indeed, the first inequality follows from the weak convergence w In particular, u ∈ W 1,p(·) B ρ , R k .
By (E.1) and (E.4), we can find a (not relabelled) subsequence and a vector m ∈ R k such that m