A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions

We prove the local H\"{o}lder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}\lambda |\nabla u|^{2}+h(\det \nabla u) \,dx\] subject to a condition of `positive twist'. The latter turns out to be equivalent to requiring that $u$ maps circles to suitably star-shaped sets. The convex function $h(s)$ grows logarithmically as $s\to 0+$, linearly as $s \to +\infty$, and satisfies $h(s)=+\infty$ if $s \leq 0$. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e\frenchspacing. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed H\"{o}lder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term $\int_{\om} h(\det \nabla u)\,dx$ can have by analysing the regularity of local minimizers in the class of `shear maps'. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are H\"{o}lder continuous.


Introduction
In this paper we consider the question of regularity of local minimizers of functionals representing the stored energy of two-dimensional elastic bodies. Ball notes in [3] that this is one of a number of outstanding open problems in the field, and while we believe that the results presented here are a positive contribution towards improving the regularity of elastic energy minimizers, we do not claim to be able to prove that the latter are smooth, which in the sense of [3, Section 2.3] would be the ideal outcome of such an effort. Instead, our goal is the more modest one of improving the regularity of continuous local minimizers to Hölder continuity.
Energy functionals in nonlinear elasticity are typically of the form where W is polyconvex, that is W (F ) is a convex function of (F, det F ), and where W (F, δ) → +∞ as δ → 0+ (1.1) for each fixed 2 × 2 matrix F with det F > 0. The constitutive condition (1.1) models the physical reality that compressing material to zero volume ought to incur an infinite energetic cost. That such a condition could be captured and embedded in an existence theory using polyconvex functions was first realized by Ball [1], whose well-known work has since given rise to a rich literature on the topic. Here, Ω is a bounded domain in R 2 with Lipschitz boundary representing the region occupied by the elastic material in a reference configuration. In circumstances where (1.1) does not hold the regularity or partial regularity of minimizers of polyconvex functionals has been studied by several authors, including but not limited to [16,13,5,4,18,17]. When (1.1) is imposed, or if the intention is to somehow faithfully approximate it, fewer results are available. These can be divided according to whether full or partial regularity is proven. The work [10] is of the former type and focuses on planar stored-energy functions of the form W (F ) = g(|F | 2 ) + h(det F ) det F ≥ 0, (1.2) where g, h : [0, +∞) → [0, +∞) are of class C 1 . Crucially, g and h are finite, meaning in particular that (1.1) cannot hold; rather, the extended real valued problem is approximated by not specifying h(0), which could therefore be as large as desired. Under suitable growth and other assumptions, including that u ∈ W 1,2 (Ω, R 2 ) has finite energy E(u) and the so-called energy-momentum equations div (∇u) T D F W (∇u) − W (∇u)1 = 0 in D ′ (Ω) hold, u is shown to be Hölder continuous, with Hölder exponent depending on the parameters appearing in the growth condition. Fuchs and Seregin point out in [10,Remark 2] that the same result can be obtained when h(0) = +∞ provided the assumptions on u are extended to include det ∇u > 0 a.e. in Ω and Ω (det ∇u) −(1+ǫ) dx < +∞ for some ǫ > 0. Unfortunately, the stored-energy functionals we consider in this paper violate the growth conditions given in [10], so that even if the condition (det ∇u) −1 ∈ L 1+ǫ (Ω) were to hold-and there seems to be no a priori way to tell whether it does or not-the proof given in [10] does not apply.
In [11] and [12], the partial regularity of minimizers of functionals with integrands typified by (1.2) is proven. Moreover, a sequence of such minimizers is used to strongly approximate in a suitable Sobolev norm a minimizer, u say, of a functional whose stored-energy satisfies (1.1). The process does not confer partial regularity on the limiting minimizer u.
More recently, Foss [9] uses a blow-up technique to establish a partial regularity result for minimizers u, say, of functionals in which (1.1) is operational. For his result to hold u is required to satisfy an equiintegrability condition, referred to as (REP), which is phrased in terms of an excess quantity. Working in W 1,p (Ω, R 2 ), with p > 8 and Ω ⊂ R 2 , and by slightly corrupting Foss's notation, define for x 0 ∈ Ω, R > 0 and M ⊂ B(x 0 , R/2), the 'excess' The term involving the determinant reflects the choice made for the function h in [9], namely h(s) = s −2 for s > 0. Property (REP) is then that for some L ∞ function Q and for each β > 0 the inequality U (x 0 , R; M ) ≤ βU (x 0 , R; B(x 0 , R))Q((∇u) x0,R ) holds whenever both U (x 0 , R; B(x 0 , R)) and |M |/|B(x 0 , R)| are sufficiently small. With this in force, u is shown to be C 1,α on an open set of full measure in Ω. It is not known whether property (REP) actually holds for minimizers of storedenergy functionals. We note that the minimizer in [9] is automatically Hölder continuous on Ω by Sobolev embedding in conjunction with the assumption p > 8.
The proposal we make in this paper is to prove local Hölder regularity by replacing the technical condition (REP) with another, simpler condition which admits a straightforward geometric interpretation. The aim is distinct from the works on partial regularity cited above in the sense that we prove a lesser degree of regularity but on a larger set, and in a situation where (1.1) is in operation. To describe the condition we impose we first fix the class of stored-energy functions to which these new arguments apply. For λ > 0 let W (F ) = λ|F | 2 + h(det F ) defined on 2 × 2 matrices F , where, for fixed constants 0 < c 1 < 1 < c 2 < +∞ and l, m > 0, h : R → [0, +∞] is given by The function θ is chosen so that h is convex and of class C 1 . We consider maps such that E(u) = Ω W (∇u) dx is finite, which, in view of the definition of h, immediately implies that det ∇u > 0 a.e. in Ω. By [22], or byŠverák's well-known regularity result [21,Theorem 5], such u are in particular continuous. Thus in our case the improvement in regularity, when it occurs, is from continuous to Hölder continuous. The condition we use to ensure the improvement is based on the nonnegativity of a quantity which for each u ∈ W 1,2 (Ω, R 2 )∩C 0 (Ω, R 2 ) is defined a.e. on a subset of Ω × Ω. We shall later refer to t(x, x 0 , u) as the twist of u at x relative to x 0 . For smooth diffeomorphisms v : R 2 → R 2 , say, with det ∇v > 0 it is easy to show that In particular, t(x, x 0 , v) > 0 for all x sufficiently close to x 0 . For a general u belonging to W 1,2 (Ω, R 2 ) ∩ C 0 (Ω, R 2 ) the same need not be true and must instead be hypothesized. For a given u we suppose that there is z ∈ Ω and δ > 0, r ′ > 0 such that t(x, x 0 , u) ≥ 0 for a.e. x ∈ B(x 0 , r ′ ) and a.e. x 0 ∈ B(z, δ). (1.3) Suppose we fix x 0 in B(z, δ) for which t(x, x 0 , u) ≥ 0 for a.e. x in B(x 0 , r ′ ). It is shown in Section 2 that this condition is equivalent to requiring that u be locally star-shaped in the sense that for a.e. R ∈ (0, r ′ ) the image u(S(x 0 , R)) of a circle S(x 0 , R) centred at x 0 and of radius R is star-shaped: see Definition 2.4. To do so we rely on the powerful technical machinery of Müller and Spector [20]. The key point is that the interaction of u(S(x 0 , R)) with rays emanating from u(x 0 ) can be understood using properties of the degree. Further details can be found in Section 2. When condition (1.3) holds we show in Section 3 that local minimizers of the functional E are Hölder continuous on compact subsets of B(z, δ). Here, u is a (strong) local minimizer if E(v) ≥ E(u) for all v such that E(v) < +∞ and ||v − u|| ∞;Ω is sufficiently small. The argument uses t(x, x 0 , u) ≥ 0 in two essential ways, by (i) ensuring that a suitable class of outer variations u ǫ = u + ǫϕ satisfies E(u ǫ ) < +∞ for all sufficiently small and negative ǫ, and (ii) establishing that the limit holds and results in a variational inequality to which elliptic regularity theory can be applied.
We remark that the logarithmic growth of h(det F ) as det F → 0+ plays a pivotal role in (ii); however, it may in future be possible to generalize the technique to other types of singularity. Indeed, in Section 4 we do this, albeit in a restricted class of competing functions. Section 4 focuses on an application of some of the ideas of Section 3 to a particular class of maps-the so-called shear maps. These allow us to treat a much wider class of stored-energy functions in the sense that the h(det ∇u) term of the stored-energy function W is required to obey much weaker growth conditions: see hypotheses (H0)-(H4) of Section 4 for details. The term h(det ∇u) has a regularizing effect on local minimizers, or at least its presence is not an impediment to the improvement of regularity. The gain in regularity is tied to imposing a condition (see (4.3)) which allows us to construct outer variations as before, and subsequently to adapting and applying a version of elliptic regularity theory.
The organisation of the paper is as follows. After introducing notation in Section 1.1, we give in Section 2 the geometric characterisation of the condition t(x, x 0 , u) ≥ 0. Section 3 is broken into three subsections, the first and shortest of which introduces the main functionals to be studied and ends with a brief proof of the existence of at least one energy minimizing deformation. The rest of Section 3 consists in establishing a variational inequality (Section 3.1) to which an adapted elliptic regularity theory applies (Section 3.2). The paper concludes in Section 4 with an analysis of the regularity of local minimizers in the class of shear maps, followed by a short appendix containing two ancillary results.

Notation
The standard notation B(x, r) for a ball of radius r and centre x will be used, and its boundary ∂B(x, r) will be written S(x, r). The terms null set and a.e. will refer to L 2 measure; in all other cases the relevant measure, most often H 1 , will be explicitly referred to through 'H 1 −null' and H 1 −a.e. respectively. Rot(ψ) will denote the matrix representing rotation anticlockwise through ψ radians, with the particular letter J reserved for Rot(π/2). The unit vector e(θ) is defined to be e(θ) := (cos θ, sin θ) T . Our notation for norms in Lebesgue and Sobolev spaces, as well as for the spaces themselves, is conventional. For clarity, we write ||u|| p;Ω for the L p norm ( Ω |u| p dx) 1/p and ||u|| 1,p;Ω for the Sobolev norm ( Ω |u| p + |∇u| p dx) 1/p when 1 ≤ p < +∞, with the usual adjustments for the case p = +∞. Here, Ω is a domain in R 2 and ∇u is the weak derivative of u. In the case of planar maps u : Ω → R 2 we write where u(x) = (u 1 (x), u 2 (x)) T and u i,j := ∂u i /∂x j . Other notation is introduced as and when it is needed.

Positive twist, condition (INV) and local starshapedness
In this section we frame the conditions needed to derive the main regularity results of the paper. Let u 0 be a homeomorphism of Ω onto u 0 (Ω) and assume that there is at least one mapping u in the class Maps belonging to A 1 will be referred to as admissible maps. To each such map and each point x 0 ∈ Ω we associate a map x → t(x, x 0 , u), which we call the relative twist of u about x 0 , and which is defined in terms of a general element of W 1,1 (Ω, R 2 ) as follows: Notice that t(x, x 0 , Ω) is only defined for a.e. x in Ω (since the same is true of ∇u). However, by convention, we shall refer to t(x, x 0 , u) as a function rather than an equivalence class. Also, although the twist is defined relative to a point x 0 in Ω and so, strictly speaking, should be referred to as a relative twist, we shall nevertheless refer to it simply as twist, hoping that no confusion arises.
We postpone until Section 3 both the motivation for studying the relative twist of an admissible map and an explanation for requiring that it be nonnegative on an appropriate subset of Ω × Ω. Instead, we focus here on the geometric consequences of requiring that the twist be locally nonnegative. To be more specific, it turns out that requiring t(x, x 0 , u) ≥ 0 for a.e. x ∈ B(x 0 , r ′ ) and for some x 0 ∈ Ω, where r ′ < dist (x 0 , ∂Ω), is equivalent to the statement that u maps circles S(x 0 , R) to star-shaped sets for a.e. R ∈ (0, r ′ ). To prove it we exploit the fact that admissible maps automatically have a continuous representative (which we henceforth identify with the map itself), which in turn allows us to apply some of the machinery of [20]. The following technical result records some of the properties of maps in A 1 .
Proposition 2.2. Let u ∈ A 1 as defined in (2.1). Then (i) u can be identified with its continuous representative; (ii) u has the N −property; where R 0 := dist (x 0 , ∂Ω), and (iii): The hypothesis u| ∂Ω = u 0 | ∂Ω , where u 0 is a homeomorphism, means that [21, Lemma 5 (i)] applies directly. Alternatively, apply (v) and [20,Lemma 3.4] to reach the same conclusion. We also present a simple, direct proof, as follows. Firstly, by [8,Theorem 5.21], the set is a set of full measure in Ω. By excluding a further subset of measure zero on which det ∇u = 0, and then if necessary relabelling Ω 1 , we can assume that det ∇u(x) > 0 for all x ∈ Ω 1 . Part (i) and [8,Lemma 5.9] then imply that for each x belonging to Ω 1 and for all sufficiently small R > 0, u(x + h) = u(x) for all h ∈ B(0, R) and that d(u, B(x, R), u(x)) = sgn det ∇u(x) = 1. The former ensures that u(x) / ∈ u(S(x, R)) and hence that the degree as written is welldefined. Let y ∈ u 0 (Ω) \ u 0 (∂Ω). Since u agrees with the homeomorphism u 0 on ∂Ω then standard properties of the degree mean that d(u, Ω, y) = d(u 0 , Ω, y) = 1, and hence that u −1 (y) is nonempty. Suppose for a contradiction that u −1 (y)∩ Ω 1 contains at least two points x 1 = x 2 . By the above, there are balls To verify it we must show that for each x 0 ∈ Ω there exists an L 1 null set N 0 such that, for all r ∈ (0, R 0 ) \ N 0 , u| S(x0,R) is continuous, By (i), the continuity of u on S(x 0 , R) for all R ∈ (0, R 0 ) is assured, so the first part is automatically true (with N 0 empty). Condition (I) holds by appealing to [21, Theorem 3, Cor. 1 (ii)], where, in their notation (and in view of the constraint det ∇u > 0 a.e.) the set E(u, B(x 0 , R)) replaces im T (u, B(x 0 , R)).
To see that condition (II) holds it is sufficient to show that the set Ω 2 := {x ∈ Ω\B(x 0 , R) : u(x) ∈ im T (u, B(x 0 , R))} has L 2 measure zero. Let x ∈ Ω 2 . Then x / ∈ S := S(x 0 , R) and so, if y := u(x), the degree d(y) := deg(u, S, y) is well defined and, by hypothesis, satisfies d(y) = 1. By properties of the degree there is at least one x ′ ∈ B(x 0 , R) such that u(x ′ ) = y, where x ′ = x in particular. Since u is 1 − 1 a.e by (iii), and in the notation of that part of the proof, we must have x ∈ Ω \ Ω 1 , which, since the latter set is L 2 null, concludes the proof of (v).
We now turn to the derivation of necessary and sufficient conditions for the local nonnegativity of the twist t(x, x 0 , u) which apply in our setting. To begin with we note that, for sufficiently regular maps u, the condition det ∇u > 0 implies that t is positive everywhere in a sufficiently small ball about x 0 . For later use we record the following result, whose proof is straightforward and is therefore omitted. Proposition 2.3. Let x 0 ∈ Ω and let u : Ω → R 2 be a diffeomorphism in a neighbourhood of x 0 . Then It follows that if det ∇u(x 0 ) > 0 then the twist of u relative to x 0 is necessarily positive on a sufficiently small ball around x 0 . The challenge is to extend this result to maps u belonging, for example, to A 1 , which clearly need not be C 1 and where det ∇u(x 0 ) need not be defined pointwise. In such cases it is possible to construct maps for which t(x, x 0 , u) < 0 for x belonging to a set of positive measure. Rather than give these examples we prefer to avoid them altogether by appealing to Lemma 2.5 below, which is phrased in terms of local star-shapedness: Definition 2.4. Let u : Ω → R 2 be continuous and let S(x 0 , R) ⊂ Ω. Then u(S(x 0 , R)) is star-shaped with respect to u(x 0 ) if: We further assume that ∇u coincides almost everywhere with its approximate derivative ap Du (see [7, Section 6.1.3], for example), and by a slight abuse of notation continue to denote the latter by ∇u. Thus, in the notation of [20, Eq. (3.14)], the induced normal on u(S) is given bỹ Note that since det ∇u > 0 a.e., |cof ∇u(x)ν(x)| = 0 a.e. in Ω. In terms of t, we have Roughly speaking, the rightmost inequality in (2.5) says that the curve u(S) turns monotonically while ν(x) traverses a circle. By Proposition 2.2 and [20, Lemma 3.5 (ii)], we may assume that the degree d(y) := deg(u, S, y) satisfies d(y) ∈ {0, 1} for all y ∈ R 2 \ u(S). Thus, in the notation of [20, Lemma 3.5], the set U 1 := {y ∈ R 2 \ u(S) : d(y) ≥ 1} coincides with the topological image im T (u, B(x 0 , R)). In particular, by [20, Step 6, Lemma 3.5],ν(u(x)) coincides H 1 −a.e. with the generalized exterior normal on the set ∂ * U 1 = u(S). Note that the latter holds up to a set of H 1 −measure zero: this follows from [20, Step 3, Lemma 3.5], where it is shown that U 1 is a set of finite perimeter (which itself follows from the fact that the degree d is a BV function), so that the reduced boundary ∂ * U 1 differs from ∂U 1 only by an H 1 −null set. To conclude the preliminaries we relate the generalized exterior normal to the tangent to u(S) as follows. Firstly, by writing for x = x 0 , we find that, by a slight abuse of notation , and e(θ) = (cos θ, sin θ) T . Thus t has a representation in local polar coordinates. Further, it is clear from Thus the generalized exterior normal to U 1 = u(B(x 0 , R)) at u(x) is obtained by scaling and then rotating u τ (x) clockwise through π/2 radians at H 1 −a.e. x in S.
To see that part (i) of Definition 2.4 holds we recall that for each By continuity, it follows that u(x 0 ) / ∈ u(S(x 0 , R)) for all R ∈ (0, δ 0 ). Therefore part (i) of Definition 2.4 holds. To prove that part (ii) of Definition 2.4 holds we make use of the assumption t ≥ 0 in conjunction with some topological observations. To begin with, by replacing u(x) with u(x) − a, we may assume that a = 0, and hence that t(x) = u(x) · Ju τ (x). Define the functions ρ(θ) and σ(θ) by Note that σ is so far only defined up to a multiple of 2π. To fix one particular σ we argue as follows. Firstly, since part (i) of Definition 2.4 holds, we can suppose that ρ(θ) ≥ c for some constant c > 0 and for all θ ∈ [0, 2π). Further, since θ → u(Re(θ)) belongs to W 1,2 ([0, 2π), R 2 ), it follows that ρ also belongs to that class. Next, note that (2.7) together with the continuity of both u and ρ implies that σ is locally continuous, for example by viewing it as suitably interpreted. It follows that if we fix σ in a neighbourhood of some θ = 0 and extend this representative to [0, 2π) then the resulting function, again denoted by σ, is uniquely defined. It is now easy to see that σ belongs to W 1,2 ([0, 2π), R), and that In particular, since t ≥ 0 a.e. by assumption, it follows thatσ ≥ 0 a.e. Now suppose for a contradiction that part (ii) of Definition (2.4) does not hold. Then there is We claim that this implies that σ strictly decreases somewhere. The proof is broken into three steps, where we make use of the more concise notation Step 1 Since u(S) is compact and does not contain 0, we can without loss of generality assume x ∈ S is such that 0 The supposition above means that there is at least one other component Γ y , say, of u(S) ∩ L u(x) containing u(y) such that Γ x ∩ Γ y = ∅ and dist (Γ x , Γ y ) is minimal. For definiteness, and by changing y if necessary, we can assume u(y) is the closest point in Γ y to Γ x .
There are two possibilities for the behaviour of the curve u(S) in a neighbourhood of u(x): either there is a branch of u(S) containing u(x) and lying in the open sector which borders and, for small ǫ 0 > 0, lies anticlockwise relative to L u(x) , or there is a branch with the same properties lying instead in the open sector lying clockwise relative to L u(x) . The second of these is eliminated by the assumption t ≥ 0 using the same argument as is given at the beginning of Step 2 below. Therefore we work with the first possibility now and make a remark in Step 2 about the impossibility of the second. By rotating L u(x) anticlockwise we generate two continuous paths in u(S) starting at u(x) and u(y), denoted by P (x) and P (y) respectively, whose construction is given below. Using the Jordan separation theorem, we write the complement of u(S) in R 2 as a union of connected, open sets, G i , i = 0, 1, 2, . . ., which we refer to as components. Since u(S) is compact there is just one unbounded component, G 0 , say. Let G 1 be the (bounded) component containing 0.
To define P (x) we proceed as follows. For ǫ 0 > 0 sufficiently small, there is a unique branch P 0 , say, of u(S) containing u(x), lying strictly in the open sector C u(x),ǫ0 and which, in addition satisfies, H 1 (∂G 1 ∩ P 0 ) > 0. Necessarily, points in P 0 are in 1 − 1 correspondence with angles of rotation ψ in the sense that each set P 0 ∩ L Rot(ψ)u(x) is a singleton for 0 ≤ ψ < ǫ 0 .
Step 2 Let Q be a branch of u(S) with initial point u(w) ∈ Γ, u(w) = u(x 1 ), and assume for a contradiction that Q lies in the open sector C u(x1),−ǫ1 for all sufficiently small ǫ 1 > 0. Recall that C u(x1),−ǫ1 borders and lies clockwise relative to L u(x1) . If there is just one branch Q as described then both Q and P 0 have a non-trivial intersection with the boundary of G 1 . The relation (2.6) then forces an orientation on both Q and P 0 , as shown in Figure 1.
In view of its inclusion in C u(x1),−ǫ1 , it is clear that σ strictly decreases along Q, contradictingσ ≥ 0. If there are two or more branches then we can find a whose boundary ∂G ′ has a nontrivial intersection with Q 1 and Q 2 , say. If d| G ′ = 0 then the orientation implied by (2.6) is shown in Figure 2 ; if d| G ′ = 1 then the arrangement is as per Figure 3 . In the first case, σ is clearly decreasing along Q 1 , while in the second σ decreases along Q 2 . Either way, we contradictσ ≥ 0. Note that this argument establishes that Γ x1 ⊂ ∂G 1 and, moreover, it shows that no branch of u(S) can lie in an open sector C u(x),−ǫ0 , as defined in Step 1.
Define u(x 2 ) as that point in u(S) ∩ Γ x1 such that |u(x 2 )| ≤ |z| for all z ∈ Γ x1 . Adjoin the interval [u(x 2 ), u(x 1 )] to P 0 and call the resulting curve P 1 . By the previous step, branches of u(S) must leave Γ x1 by entering a cone of the form C u(x1),ǫ2 , where ǫ 2 > 0. Moreover, since u : S → u(S) is 1 − 1 H 1 −a.e., at least one of these branches must contain u(x 2 ). Therefore P 1 can be extended continuously from u(x 2 ) by following the branch of u(S) which pointwise minimizes its distance to 0. Iterating this process produces the desired curve P (x), which by construction is contained in ∂G 1 . The method used to define P (y) is so similar that we omit the details, apart from saying that the condition P (y) ⊂ ∂G 1 is not required to hold.
Step 3 Since u(S) is connected it must be that P (x) and P (y) meet at some point u(w), say. Let u(w) be the first such meeting point (relative to x and y) in the sense that there is ǫ 0 > 0 such that For definiteness, let P (x) and P (y) both terminate at u(w). A glance at Figure  4 may help to visualize the arrangement.
The degree d is of compact support and is constant on each component, so it must in particular be that d| G0 = 0 and d| G1 = 1, where G 0 and G 1 were defined at the outset of Step 1. Let G be the component of R 2 \u(S) such that for all sufficiently small γ the sets G ∩ B(u(w), γ) ∩ P (x) and G ∩ B(u(w), γ) ∩ P (y) are nonempty. The boundary of each component is contained in u(S), which is of finite perimeter, so ∂G is in particular H 1 measurable (and of finite H 1 measure). By construction H 1 (∂G ∩ ∂G 1 ) > 0, so that necessarily d| G = 0. Moreover, H 1 (∂G ∩ P (y)) > 0 together with d| G = 0 implies there is a further component G ′ = G 0 such that d| G ′ = 1 and for which u(w) ∈ ∂G ′ . The local behaviour of the degree together with (2.6) forces an orientation on the set u(S) in a neighbourhood of u(w), as shown in Fig 4. Choosing Figure 4: Topological representation of the components of R 2 \ u(S) near the first meeting point u(w) of P (x) and P (y). The notation G | n is used to indicate that the degree d is n on the component G.
(u(S) is star-shaped ⇒ t ≥ 0). Since part (i) of Definition 2.4 holds we may assume that σ has been chosen so that (2.7) and (2.8) apply. Thus it is sufficient to show thatσ ≥ 0 a.e. in [0, 2π). Suppose for a contradiction thatσ ≥ 0 does not hold a.e. in [0, 2π). By reparametrizing we can suppose that 0 is a Lebesgue point ofσ and thatσ(0) < 0. Then σ(θ) < σ(0) if θ is sufficiently small and positive. Now, either σ is nonincreasing on the entire interval [0, 2π) or there is θ 1 > 0 such that σ(θ) > σ(θ 1 ) if θ −θ 1 is sufficiently small and positive. Without loss of generality we can suppose that θ 1 is the first point in (0, 2π) at which σ fails to be nonincreasing. In particular, for each sufficiently small h there is k(h) such that the sets are both contained in the same line. According to part (ii) of Definition 2.4, the sets L u(x) ∩ u(S) are connected for H 1 −a.e. x ∈ S. But this implies that u fails to be 1 − 1 on a set of positive H 1 measure, which contradicts part (iv) of Proposition 2.2. Thus the only possibility is that σ is nonincreasing on the whole interval [0, 2π). But then the set u(S) must be traversed clockwise with increasing θ, and hence by (2.6), the generalized exterior normal points almost everywhere into the set u(B(x 0 , R)), which implies d| G1 = 0, a contradiction. Thus we conclude thatσ ≥ 0 must hold almost everywhere.

A variational inequality, positive twist and Hölder regularity
In this section we introduce two functionals, E(u) and F (u, Ω ′ , r ′ ): the first measures the elastic stored energy of a deformation u belonging to the class A 1 as defined in (2.1), while the second quantity has the key property that maps u with F (u, Ω ′ , r ′ ) = 0 have a.e. positive twist on balls of radius r ′ centred at points of Ω ′ . In view of the characterization of positive twist given in Section 2, F (u, Ω ′ , r ′ ) = 0 encodes the condition that u maps these balls to star-shaped sets on a 'small scale'. We regard F (u, Ω ′ , r ′ ) = 0 as a condition which may or may not be satisfied rather than as a constraint to which all competing functions in A 1 are subjected 1 . The main result is that any local minimizer of E such that F (u, B(z, δ), r ′ ) = 0 is Hölder continuous on any compact subset of B(z, δ) ⊂ Ω. Thus, when it holds, the geometric condition involving star-shapedness translates into a means for proving regularity of the associated local minimizer of the energy E. It does this by ensuring that (i) a certain useful class of variations is admissible, and (ii) the resulting variational inequality is suitably monotone in the sense that elliptic regularity theory can be applied to it. Details of (i) and (ii) are given later in the section.
In the region s > 0 the main features of h(s) are its logarithmic growth for s small and positive, and linear growth for s large and positive. It is straightforward to check that when (i)-(v) hold the function h is C 1 and convex, with a unique global minimum of θ(1) at 1. The connecting function θ is of lesser importance, and there are many of these for which (i)-(v) hold. For example, let θ 1 be a given constant satisfying θ 1 > m + l and define The positive, continuous functions ψ 1 and ψ 2 are subject to which can be satisfied by choosing ψ 1 and ψ 2 to be appropriate polynomials.
We now define the class of admissible deformations by The functional F(u, Ω ′ , r ′ ). As advertised, F will encode our notion of 'positive twist'. To motivate its construction, suppose for argument's sake that A contains at least one diffeomorphism, v, say, where, in particular, det ∇v > 0 in Ω. By Proposition 2.3, we can assert that for each x 0 in Ω there is some radius r(x 0 ) > 0 such that Here, r(x 0 ) < dist (x 0 , ∂Ω); note also that because v is C 1 we can suppose r(·) is bounded uniformly below on compact subsets Ω ′ , say, of Ω. Let the constant r ′ satisfy 0 < r ′ ≤ inf{r(x 0 ) : x 0 ∈ Ω ′ }. Now let the function g : R → [0, +∞) be given by In view of the inequality satisfied by t(x, x 0 , v) above, it is clear that The definition of F below generalizes this idea to other admissible functions u. Thus to each u ∈ A 1 , each set Ω ′ which is compactly contained in Ω and each fixed r ′ ∈ (0, dist (Ω ′ , ∂Ω)) we associate the functional It can easily be checked that g(t(x, x 0 , u)) is integrable on the set indicated, and that F (u, Ω ′ , r ′ ) ≥ 0.
In the next section we will consider local minimizers of the functional E. To prove that at least one of these exists we appeal to the following well-known result concerning the existence of a global minimizer of E in A. Since any global minimizer is in particular a local minimizer, the result clearly establishes the existence of the latter.

A variational inequality
We begin by introducing a useful class of outer variations {u(x; x 0 , ǫ) : ǫ < 0, x 0 ∈ Ω} about a given u in A. The variations themselves are standard from the point of view of classical regularity theory in that they are of the form where ǫ is a small parameter, the scalar function η(·; x 0 ) takes values in [0, 1] and is supported in a small ball about a given x 0 in Ω. The technical distinction needed in the elasticity setting is that we also require u(x; x 0 , ǫ) to belong to A 1 , which necessarily implies that det ∇ x u(x; x 0 , ǫ) > 0 must hold almost everywhere in Ω. The latter can be achieved by imposing on u a condition of the form F (u, B(z, δ), r ′ ) = 0, where B(z, r) ⊂ Ω, and by choosing the support of η to be sufficiently small. Indeed, if t(x, x 0 , u) ≥ 0 for x ∈ B(x 0 , r ′ ) and x 0 ∈ B(z, δ), it follows from the calculations given in Proposition 3.2 that det ∇ x u(x; x 0 , ǫ) ≥ det ∇ x u(x)/4 on all of Ω provided spt η(·, x 0 ) is contained in B(x 0 , r ′ ). Hence, by properties of the stored-energy function W , E(u(·, x 0 , ǫ)) < +∞. Let us assume that F (u, B(z, δ), r ′ ) = 0 (3.6) for some ball B(z, δ) and some r ′ > 0. It follows that t(x, x 0 , u) ≥ 0 for a.e. x ∈ B(x 0 , r ′ ) and a.e. x 0 ∈ B(z, δ).

Remark 3.3.
It is possible to show that the functional F (u, Ω ′ , r ′ ) is lower semicontinuous with respect to weak convergence in W 1,2 for fixed Ω ′ and r ′ (see Lemma A.2). In particular, if u (j) ⇀ u in W 1,2 , if F (u (j) , Ω ′ , r ′ ) = 0 for all j, where each u (j) belongs to A, and since F is nonnegative, then F (u, Ω ′ , r ′ ) = 0. The existence of a minimizer u of E in the restricted class is then, in conjunction with [2, Theorem 6.1], not difficult to show. We note that A F is nonempty provided one assumes that A 1 contains at least one diffeomorphism for which F (v, Ω ′ , r ′ ) = 0: see Proposition 2.3 and the construction of F for the details. As the results of Section 2 show, the constraint F (u, Ω ′ , r ′ ) = 0 translates into the condition that the minimizer of E in A F maps sufficiently small circles centred at points z of Ω ′ to star-shaped sets relative to u(z). The catch is that in order to exploit this minimality we require not only that u ǫ ∈ A 1 , which is assured by F (u, Ω ′ , r ′ ) = 0, but also that F (u ǫ ) = 0 in order that u ǫ ∈ A F . The latter condition does not seem to hold, and so one cannot conclude that E(u ǫ ) ≥ E(u).
We now focus on deriving a variational inequality under the assumption that the relative twist t(x, x 0 , u) is nonnegative a.e. on a ball about x 0 . As we have seen, this is certain to be the case for a.e. x 0 in B(z, δ) provided F (u, B(z, δ), r ′ ) = 0 for some r ′ > 0. In the following we use the shorthand notation  2) and assume that u ∈ A. Assume futher than t(x) ≥ 0 for a.e. x in B(x 0 , r ′ ) for some r ′ > 0. Let u ǫ be given by (3.12), where ǫ < 0. Then u ǫ ∈ A,

17)
the right-hand side of (3.17) being independent of ǫ.
Proof. The assertion that u ǫ ∈ A involves showing that u ǫ ∈ A 1 and that E(u ǫ ) is finite. The former holds easily; the latter can be checked by using (3.15)  Inequality (3.17) results from an application of (3.16) to the following: The preceding estimates are needed to calculate a bound on the quantity which appears in the variational principle set out in Theorem 3.7.
Lemma 3.5. Let E be defined by (3.1) and let u ∈ A be a strong local minimizer of E in A in the sense that there is γ > 0 such that Assume that F (u, B(z, δ), r ′ ) = 0 and that x 0 ∈ B(z, δ) is such that t(x) ≥ 0 for a.e. x ∈ B(x 0 , r ′ ). Then Proof. Let u ǫ be defined by (3.12). Since u ǫ is continuous and η is bounded, it follows that ||u ǫ − u|| ∞;Ω → 0 as ǫ → 0. Applying (3.18), we can assume that for ǫ smaller in magnitude than min{γ, 1/2}, and hence, since ǫ < 0, The derivation of follows by applying (3.13) together with a suitable dominated convergence argument.
The argument needed to derive the terms on the right-hand side of (3.19) is more delicate. One approach would be to apply a dominated convergence theorem in conjunction with an estimate such as (3.17), but this could fail because for small values of d u (x), and recalling the definition of t given in (2.2), the term on the right-hand side of (3.17) is potentially of order |(∇u) −1 |, which we cannot assume to be in L 1 loc (Ω) . However, it does apply on Ω 0 , where and where δ 0 < c 1 is a small, positive quantity to be chosen shortly. Now, by (3.20) 2λ By (3.17), Splitting the range of integration in (3.23) into (3.24) The first inequality follows from the fact that h ′ (d u ) < 0 if d u < 1 and ηtf ′ (R) ≤ 0 by construction, so −h ′ (d u )ηtf ′ (R) ≤ 0 on Ω − 0 . A similar argument using the fact that 0 ≤ h ′ (d u ) ≤ l on Ω + 0 yields the second inequality. Notice that The other term on the right-hand side of (3.22) can be dealt with as follows. (3.25) and notice that, by (3.14), We now write where The constant c 1 < 1 is introduced in (3.2), according to which if x belongs to Ω 2 (ǫ) then where the last inequality holds because ϕ ≥ 0 > ǫ, so that (ln(d u +ϕ)−ln d u )/ǫ ≤ 0, and by elementary estimates for ln(1 + ǫη 2 ). It follows that But Ω 2 (0) = {x ∈ Ω : d u (x) < min{c 1 , δ 0 }}, which, because δ 0 < c 1 , coincides with the set Ω 1 . It remains to consider the sets Ω 3 (ǫ) and Ω 4 (ǫ).
For any x ∈ Ω(ǫ, c 1 ) it holds that so that, on using the definition of ϕ given in (3.25), the fact that d u < δ 0 on Ω 1 , and the assumptions |ǫ| < 1/2 and η 2 ≤ 1, Hence By the definition of t given in (2.2) and the fact that f ′ has support in a fixed annulus about x 0 , the integrand |f ′ (R)|ηt is clearly in L 2 (Ω). The claim now follows. The next step in the proof of the proposition consists in showing that when j = 3 and 4. The argument needed in the case of Ω 3 (ǫ) is as follows. Let x ∈ Ω 3 (ǫ), so that c 1 < d u ǫ < 1. Note that lim ǫ→0 d u ǫ (x) = d u (x) < δ 0 < c 1 , and hence, sinceǫ → d uǫ is quadratic inǫ, there exists a unique ǫ 1 (x), say, measurable in x, belonging to the interval (ǫ, 0) and such that The quotient A can be estimated by writing The right-hand side of the last line above is clearly an integrable function, so that, by standard results together with the claim above, Ω3(ǫ) A(x) dx → 0 as ǫ → 0. B can be estimated similarly, this time by appealing to the expressions in (3.26) but with ǫ 1 (x) in place of ǫ, followed by an integration over Ω 3 (ǫ) and another application of the claim above. Next, we consider the integral of the difference quotient over Ω 4 (ǫ). Note that because 1 + ǫη 2 < 1 we have d u ǫ < d u + ϕ, and therefore since 1 ≤ d u ǫ and h is increasing on (1, +∞), Let δ 0 be so small that the first two integrands are negative. The term in ϕ satisfies where the integrand 4l|f ′ (R)|ηtχ Ω + 1 (ǫ) converges a.e. and boundedly in L 2 (Ω) to zero (by the claim above). We have therefore shown that To finally obtain (3.19), take the limit ǫ → 0 in (3.20) and apply (3.21), (3.23), (3.24), (3.27) and (3.28). This gives The remarks above imply that and we have already pointed out that Ω + 0 is the set on which d u (x) ≥ 1. Hence (3.29) implies (3.19) when the definition of h given in (3.2) is applied.
In summary, (3.32), together with (3.33) and (3.34), yields φ(r) ≤c(p, r 1 ) max{r for some constantc with dependence as shown. The statement of the lemma now follows.
Theorem 3.7. Let E be defined by (3.1) and suppose that u is a strong local minimizer of E in A. Suppose further that F (u, B(z, δ), r ′ ) = 0 for some B(z, δ) ⊂ Ω and r ′ > 0, where F is given by (3.4). Then u is Hölder continuous on any compact subset of B(z, δ).
Proof. By Lemma 3.5, we can assume that for almost every x 0 in B(z, δ) the inequality (3.19) holds. The right-hand side of (3.19) consists of two terms which satisfy the following estimates: The first estimate uses the definition of h given in (3.2); the second uses the fact that η(x) = f (|x − x 0 |), where f has support in [0, 2r], along with (2.2).

Hölder regular shear solutions
In this section we exhibit a variational problem from nonlinear elasticity theory to which some of the ideas described earlier in the paper apply. Our purpose is to directly illustrate that the singular term involving h(det ∇u) appearing in the energy functional has a regularizing effect on the minimizer. We consider a restricted class of deformations, which we call shear maps, and which take the form Here, σ is a scalar-valued map which belongs to a certain subclass of the space W 1,2 (Q; R) and Q is the set [−1, 1] 2 in R 2 . The choice of Q is not pivotal, but it does enable the problem to be visualised more easily. If we let l x1 = {(x 1 , t) T : t ∈ R} be the vertical line through (x 1 , 0) T then u σ maps Q ∩ l x1 to a subset of l x1 for each −1 < x 1 < 1. Thus nearby lines in Q are 'sheared' relative to one another. An advantage of using shear maps u σ is that the Jacobian det ∇u σ is affine in σ ,2 . Specifically, a short calculation shows that det ∇u σ = 1 + σ ,2 .
(4.1) Thus, in the notation introduced in Section 3, the energy E(u σ ) of a shear map takes the form The term involving h(1 + σ ,2 ) has a regularizing effect on σ in the x 2 -direction only. Therefore, it seems to be necessary to assume some extra regularity in the x 1 direction. We impose the condition that for some M which clearly amounts to assuming that σ(x 1 , x 2 ) is Lipschitz in the x 1 -direction with Lipschitz constant independent of x 2 . We conjecture that this can be weakened to a uniform Hölder assumption on σ in the x 1 direction. The form of the Jacobian (4.1) for shear maps means that we can treat a wider class of stored-energy functions than those detailed in Section 3, where we can in particular allow quadratic growth of h(s) as s → +∞ and the behaviour of h(s) as s → 0+ need not be logarithmic in s. See Section 4.1 for the assumptions applied to h in the shear map setting. The affine Jacobian (4.1) also allows us to design outer variations u σ ǫ , say, within the class of shear maps which obey the constraint det ∇u σ ǫ > 0 a.e. In previous sections of the paper this was achieved by imposing a condition of positive twist; here, we avoid imposing that condition, which is hard to verify, and replace it with the simpler condition (4.3) in the x 1 -direction.
Definition 4.1. Let u σ0 be such that E(u σ0 ) < +∞ and let (∂Q) D ⊂ ∂Q. Define the class A s of admissible shear maps as follows: Here, u σ = u σ0 on (∂Q) D in the sense of trace, and we have suppressed the dependence of A s on σ 0 and (∂Q) D .
The existence of a minimizing shear map u σ in A s now follows from methods established in [2]. Proof. By [2, Theorem 6.1], the integrand W defined in (4.4) subject to assumptions (H0)-(H3) is such that E is sequentially lower semicontinuous with respect to weak convergence in W 1,2 . Moreover, it is clear that if u σ (j) is a minimizing sequence for E in A s satisfying u σ (j) ⇀ v for some v ∈ W 1,2 (Q, R 2 ) then v must also be a shear map, that is v = u σ for some σ whose trace on (∂Q) D agrees with that of u σ0 . Hence u σ minimizes E in A s . The next result is a technical lemma which uses condition (4.3) and the determinant constraint det ∇u σ > 0 a.e. to generate one-sided bounds on σ. We use the notation x 0 = (x 01 , x 02 ) T .
for almost every x ∈ B(x 0 , r) such that x 2 < x 02 , and for almost every x ∈ B(x 0 , r) such that x 2 > x 02 .
Proof. First note that E(u σ ) < +∞ implies det ∇u σ > 0 almost everywhere. In view of (4.1), this gives σ ,2 > −1 almost everywhere. Let x 2 < x 02 . Since σ is absolutely continuous along almost all lines, we can suppose that for almost every x 0 it is the case that Hence (4.6) holds with C = 1 + M . The proof of (4.7) can be obtained by exchanging x and x 0 in the above, so that the same constant C suffices for both inequalities.
Remark 4.4. Note that (4.6), which follows from the assumption σ ,2 > −1, gives an upper but not lower bound on σ(x) − σ(x 0 ) in the region x 2 < x 02 . The lower bound will follow from an application of elliptic regularity theory, as we shall see. Similar comments apply to the inequality (4.7).
We are now in a position to form a variational inequality which in fact applies to a local minimizer of the energy E(·) in the class A s . The following result is analogous to Lemma 3.5.
Proof. Define u σ ǫ,± as in Proposition 4.5 and note that u σ ǫ,± → u σ in W 1,∞ (Q, R 2 ) as ǫ → 0. In particular, since u σ ǫ,± belong to A s and u σ is by hypothesis a local minimizer in that class, it follows from (4.14) that for all sufficiently small ǫ < 0. Let us focus on the variations u σ ǫ,+ . The quotient in (4.16) consists of two terms, the first of which obeys The second term in the quotient (4.16) essentially involves taking the derivative of the energy with respect to the singular term Q h(det u σ ǫ,+ ) dx. To do this we use ideas from the proof of Lemma 3.5, beginning by rewriting (4.16) as Let δ 0 > 0 be a small positive constant to be chosen later and define A dominated convergence argument then yields lim sup Noting that −h ′ (det ∇u σ ) ≥ 0 when det ∇u σ ≤ 1, and that σ ,2 < 0 on the same set, it follows that −h ′ (det ∇u σ )ζ(x) ≤ 0 on {x ∈ Q + 0 : det ∇u σ ≤ 1}. (Here one uses that (∇R) 2 χ + ≥ 0.) Thus the last integral above is bounded above by 0. In view of (4.19), the fact that h is increasing on [1, +∞) and since σ ,2 > 0 when det ∇u σ > 1, the penultimate integral is bounded above by |h ′ (det ∇u σ )|(|∇(η 2 ))|ξ + + Cη 2 )χ + dx.
In summary, where d := det ∇u σ and Q − 0 ∩ {d + ǫζ < 1} is shorthand for {x ∈ Q − 0 : d(x) + ǫζ(x) < 1}. Notice that d + ǫζ ≥ d on the set where d < δ 0 (since ζ(x) ≤ 0 there, as can be seen by inspecting (4.19)), and so, because h(s) is decreasing for 0 < s < 1, it follows that To estimate the second integral we first use (H3) to deduce that there is a constant q 3 such that h(s) ≤ q 1 s 2 + q 2 s + q 3 if s ≥ 1.
The fact that ζ belongs to L 2 (Q) also clearly implies that Sǫ ǫζ 2 dx tends to zero as ǫ → 0. Hence, when we recall that d ≤ δ 0 on the set over which we are integrating, (4.22) holds, from which it follows that lim sup The preceding lemma now allows us to improve the regularity of σ in the x 2 direction, as follows.
Theorem 4.7. Let E be defined by (4.5) and let u σ be a strong local minimizer of the functional E (in the sense of (4.14)) in the class A s . Assume that condition (4.3) holds. Then u σ is locally Hölder continuous in the set Q.
Proof. The proof has two steps.