On singularities of stationary isometric deformations

Unstretchable thin elastic plates such as paper can be modelled as intrinsically flat W2,2 isometric immersions from a domain in R2 into R3. In previous work it has been shown that if such an isometric immersion minimizes the elastic energy, then it is smooth away from a singular set consisting of three different subsets. In the present paper, we show that each of these singular subsets can indeed occur and that regularity may indeed fail there.


Introduction
In this article we are address the regularity properties of stationary points of the Kirchhoff plate (or 'elastic bending energy') functional from nonlinear elasticity. This is the functional on the class of isometric deformations u : S → R 3 . Here S ⊂ R 2 is a bounded domain and A is the second fundamental form of the immersion u, and by an isometric deformation we mean a W 2,2 isometric immersion, i.e. a map u in the Sobolev space W 2,2 (S, R 3 ) which satis es the constraint (∇u) T ∇u = I almost everywhere on S.
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Here I ∈ R 2×2 is the unit matrix. Such maps model elastic deformations with nite bending energy and with zero membrane energy, i.e. which neither stretch nor shear the material. For such u the second fundamental form is the matrix eld A : S → R 2×2 with entries where the dot denotes the Euclidean scalar product and the cross the cross product in R 3 . Kirchhoff's plate functional models the behaviour of inextensible thin elastic plates such as a sheet of paper. It was derived rigorously from nonlinear three dimensional elasticity in [7]. More precisely, it was shown in [7] that the functionals which associate with a deformation z : S × (0, h) → R 3 of a thin lm with thickness h the scaled elastic energy converge, in the sense of Γ convergence, to Kirchhoff's plate functional.
Kirchhoff's plate functional agrees (up to a constant prefactor) with the Willmore functional from differential geometry [25,26] restricted to the class of isometric immersions. This is because the Gauss equation asserts that det A = 0 for isometric immersions, so |A| 2 is proportional to the squared mean curvature.
A nontrivial variational problem for (1) is obtained by prescribing boundary conditions. However, due to the isometry constraint the problem is quite rigid. Therefore, boundary conditions are typically prescribed only on a (strict) closed subset ∂ c S of ∂S with nonzero length. In order to prescribe values which are compatible with the isometry constraint, one xes some isometric immersion u 0 ∈ W 2,2 (S; R 3 ) and seeks minimizers of (1) within the class of W 2,2 isometric immersions u satisfying u = u 0 und ∇u = ∇u 0 on ∂ c S.
Existence of minimizers in this setting is quite direct, see e.g. [12,14]. The Euler-Lagrange equations satis ed by minimizers are derived in [12] and, in the same paper, a regularity result is established that can be summarised as follows: outside a closed singular set the minimizer in question is C 3 and outside a larger singular set it is C ∞ .
The singular set arising in [12] consists of line segments. This is due to the fact that W 2,2 isometric immersions are developable. This means that through any point x ∈ S around which u is not planar, there exists a unique line segment with endpoints on ∂S, such that u is af ne along this segment.
More precisely, the singular set consists of essentially three families of such segments. Each family is de ned in a geometric way. And each of them arises in [12] for apparently technical reasons. So it is a priori not clear if they occur at all and if regularity actually fails there, i.e. if the regularity result from [12] is sharp.
In the present work we essentially con rm that these results are sharp. We provide explicit examples of smooth and simply connected domains S and boundary data ∂ c S and u 0 , such that the corresponding minimizing isometric immersions u display the singular set in question and fail to be more regular than asserted in [12]. Observe that the results in [11,13], in contrast, show that a 'generic' W 2,2 isometric immersion (under similar boundary conditions) is in C ∞ .
Both in the mathematical and in the applied literature there has been considerable interest in isometric deformations with singularities. For instance, conical singularities were studied in [1,2,19,27]. Another instance is folded or crumpled paper. This topic was addressed, e.g. in [4,16,18,23,24,27].
One main question in those situations is about the scaling of the three dimensional elastic energy (3) of a thin sheet with respect to its thickness h. The singularities of the asymptotic deformations are re ected by the fact that the energy of the actual deformations z will be much greater asymptotically than that of deformations with nite bending energy.
A somewhat related question is about the rigidity of isometric immersions. It is possible to construct very exible isometric deformations with continuous deformation gradient [17,20] and even with Hölder continuous deformation gradient [3,6]. Such constructions must create singularities on the level of the second fundamental form.
The singularities addressed here are different from those just discussed in two main respects. First, the admissible deformations in the present paper always have nite bending energy. This is not the case for cones, where the bending energy concentrates in a point, nor for folds, where the bending energy concentrates along folds, and even less so for convex integration solutions, which differ dramatically from deformations with nite bending energy. In particular, they fail to be developable and there is no meaningful way of de ning their second fundamental form. In contrast, the admissible deformations considered here are developable. In [23] the authors study the extent to which developability leads to a lower bound on the extrinsic diameter of the deformed con guration. Such a bound shows that they are fundamentally different from the exible isometric immersions in [17,20].
Our work is related to [7], in that the setting is precisely the asymptotic zero thickness theory-Kirchhoff's plate theory-derived in that article.
The second difference is that the deformations constructed here, in addition to having nite bending energy, are minimizers of Kirchhoff's plate functional (1).
Finally, notice that our results depend entirely on the developability of the admissible deformations and therefore differ from situations in which the prescribed metric is not intrinsically at. However, somewhat related observations to ours are made in [8,9] for the case of hyperbolic metrics. There it is shown that nonsmooth isometric deformations with nite bending energy may have lower energy than smooth ones.
This article is organised as follows. In the remainder of this Introduction we provide some more details about the regularity result in [12]. In section 2 we present some further facts about isometric deformations, including a result about failure of regularity. Finally, in section 3 we construct examples of stationary points which display the singular sets encountered in [12] and which fail to be regular. The appendix contains some more technical proofs.

Developability of isometric immersions and partial regularity
We begin by discussing some basic properties of the admissible deformations that are required to formulate the main (positive) regularity result from [12].
Throughout this article S ⊂ R 2 models the reference con guration of an (in nitely thin) inextensible elastic sheet. We take S to be a bounded, simply connected domain with boundary of class C ∞ . The admissible deformations are those with nite Kirchhoff bending energy, i.e. they belong to the set Here δ ij is the Kronecker symbol. As shown in [15], every u ∈ W 2,2 iso (S; R 3 ) belongs to C 1 (S; R 3 ). Let u be a W 2,2 isometric immersion. In [15,21,22] it is shown that then u is 'developable' away from the open set This means that through every point x ∈ S\C ∇u there exists a unique line segment [x] ⊂ S with both endpoints on the boundary ∂S of S such that the deformation gradient ∇u is constant on [x]. As a result the image of [x] under the deformation u will still be a straight segment. We refer to [10] for a classical version of this developability result. Developability will be essential throughout this article.
We will next state the regularity result proven in [12]. For a given u 0 ∈ W 2,2 iso (S; R 3 ) and a closed subset ∂ c S ⊂ ∂S with positive length, we de ne the class The equality ∇u = ∇u 0 is understood in the trace sense. These are the deformations satisfying on ∂ c S the clamped boundary conditions determined by u 0 . We are interested in the minimizers of (1) within the class A u 0 (S, ∂ c S). The existence of such minimizers is easy to show, cf [12] or, for a more general context, [14]. From now on the word 'minimizer' will always refer to a minimizer of this kind. The singular set encountered in [12] consists of the following subsets intersects ∂S tangentially at one or both end-points} , as well as the relative boundary S ∩ ∂ C ∇u of the set C ∇u =union of all connected components U of C ∇u whose relative boundary S ∩ ∂U consists of at least three connected components.
Later we will see that the connected components of S ∩ ∂U are segments of the form [x]. By ( [13], proposition 14) the set Σ τ is relatively closed in S. Another set arising naturally in [12] is The regularity result ( [12], theorem 1.3) is the following: Theorem 1.1. Let S ⊂ R 2 be a bounded C ∞ -domain, let ∂ c S ⊂ ∂S be closed and let u be a minimizer of (1) within A u 0 (S, ∂ c S). Then and The singular sets in (6) can be visualised as follows: the set denoted by Σ c consists of those segments on which u is determined (due to it being af ne along those segments) by the boundary conditions prescribed on ∂ c S. The set Σ τ consists of those segments which intersect the boundary ∂S tangentially. And the set S ∩ ∂ C ∇u consists of those segments which constitute the boundary (in S) of certain regions on which u is planar. In section 2.3 we recall the relevant facts from [5,13] about the shape of these planar regions. Essentially, they are polygons with vertices on the boundary of S. Figure 1 depicts this description.

Failure of regularity
As mentioned earlier, in section 3 we will provide examples showing that theorem 1.1 is essentially optimal, in the sense that one can nd appropriate data S, ∂ c S and u 0 such that minimizers of (1) within the corresponding class A u 0 (S, ∂ c S) of admissible deformations actually have nonempty sets Σ c , Σ τ or C ∇u and indeed fail to be C ∞ across these sets. It is clear that the set Σ c in general will be an obstruction to regularity since the boundary datum u 0 is not assumed to be regular. In section 3.2 we spell out a straightforward example regarding this simple singular set Σ c .
Regarding the set C ∇u , notice that it may well be empty in general. In fact, in [5] it was shown that it will be empty unless ∂ c S consists of at least three subintervals of the boundary.
In section 3.3 we provide an explicit and intuitive example which shows that, under suitable (and realistic) boundary conditions, minimizers must have a planar region U bounded by at least three straight line segments (therefore U ⊂ C ∇u ), and that u fails to be smooth at each of these segments.
If the domain S is convex, the set Σ τ is clearly empty. In section 3.4 we give an example showing that, if the reference domain S is nonconvex, then minimizers can have a nonempty set Σ τ and that they fail to be smooth.

Preliminaries
In this section we provide some technical results that will be essential in the construction of the examples mentioned earlier. In section 2.1 we prove some simple but useful properties related to the metric properties of isometric immersions. In section 2.3 we recall some facts about the set C ∇u and its connected components. The main non-regularity result is stated in section 2.4.

Metric lemmas
By de nition we have u = u 0 and ∇u = ∇u 0 on ∂ c S for all u ∈ A u 0 (S, ∂ c S). Due to the isometry constraint, these boundary conditions may determine u in the interior of S as well. |u(z) − u( y)| |z − y| for all z, y ∈ l, and assume that Then u is an af ne isometry on l, i.e.
Proof. Let x ∈ l be a point on the line (ab). Then, since a, x and b are parallel, |b − x| + |x − a| = |b − a|. The triangle inequality, combined with the assumptions of the lemma, give Since the leftmost and rightmost terms agree, both inequalities must in fact be equalities. Together with the Lipschitz condition, the second (in)equality implies that |u(b) − u(x)| = |b − x| and |u(x) − u(a)| = |x − a|. Then by the triangle inequality, we get for any y, z ∈ l: Together with the Lipschitz condition, the assertion follows.
This sort of metric results is relevant in the context of the present paper because clearly every u ∈ W 2,2 iso (S) belongs to C 0 (S, R 3 ) and is 1-Lipschitz on S. An application of the previous lemma therefore yields the following: (This is the case if x 1 and x 2 belong to a segment of constancy of u 0 , cf below.) If u ∈ W 2,2 iso (S) agrees with u 0 in x 1 and x 2 , then in fact u = u 0 on the whole segment with endpoints x 1 and The next lemma is essentially a two-dimensional version of lemma 2.1. In what follows, B r (x) denotes the open disk of radius r centred at x. Lemma 2.3. Let u : B r (x) → R 3 be 1-Lipschitz and let a 1 , b 1 , a 2 , b 2 ∈ ∂B r (x) such that the segments (a 1 b 1 ) and (a 2 b 2 ) have exactly the point x in common. In addition, let u be an af ne isometry on (a 1 b 1 ) and on (a 2 b 2 ). Then u is an af ne isometry on the whole convex hull conv{a 1 , Proof. Without loss of generality we assume x = 0 and u(0) = 0 Then the isometry condition states that |u( We will now show that that is, u preserves the angle between (a 1 b 1 ) and (a 2 b 2 ), or (0b 1 ) and (0b 2 ). Indeed, suppose that equation is wrong, say, Then we can calculate but this contradicts the Lipschitz-condition. Now let y 1 ∈ (0b 1 ) and y 2 ∈ (0b 2 ). The previous lemma shows that |u( y i )| = |y i |. With a similar calculation to the above, we can show that u is af ne on ( y 1 y 2 ): With analogous inferences for y 1 ∈ (0a 1 ) and so on, we nally conclude that u is af ne on conv{a 1 , Then either l ⊂ C ∇u or l ⊂ S\C ∇u . In the latter case, l ⊂ [x] for any x ∈ l.
Proof. Assume that l is not contained in C ∇u and let x ∈ l\C ∇u . Then [x] is well-de ned and is contained in S\C ∇u . If [x] were not parallel to l, then lemma 2.3 would imply that x ∈ C ∇u , a contradiction. Hence l is parallel to [x] and therefore l ⊂ [x] by maximality of [x].
Recall that W 2,2 iso (S) embeds into C 1 (S), so pointwise values of ∇u are well-de ned. Definition 2.5. A segment l ⊂ S is said to be a segment of constancy of ∇u if both endpoints of l lie on ∂S and ∇u is constant on l. Remark 2.6. Let u ∈ W 2,2 iso (S). Then the following are true: Proof. The rst assertion is in fact part of our de nition of developability. The second one follows from ( [13], proposition 1) and ( [13], proposition 9). Remark 2.7. Let u ∈ W 2,2 iso (S) and let l ⊂ S be a nondegenerate segment with endpoints on ∂S. Then the following are equivalent: (a) The segment l is a segment of constancy of ∇u.
Proof. If l is a segment of constancy, then clearly u is an af ne isometry on l. Conversely, if |u(x 1 ) − u(x 2 )| = |x 1 − x 2 | then lemma 2.4 implies that l is a segment of constancy.

Developability
In the language of [11,13], the developability of maps u ∈ W 2,2 iso (S; R 3 ) can be stated as follows: there is a ruling for ∇u, i.e. a map q : x] e and whose direction is given by e ∈ R 2 . The ruling q is unique (after identifying antipodal points). Moreover, for all . This condition implies that q is Lipschitz on compact subsets of S\C ∇u if regarded as a mapping into the projective space P 1 , with Lipschitz constant near x dominated by (dist(x, ∂S)) −1 [see ( [13], remark 1 p 959) and ( [15], proposition 2.30(i))].
In what follows we will write [x] instead of [x] q(x) unless there is a danger of confusion. In cases where it is not clear which immersion u ∈ W 2,2 iso (S, R 3 ) the ruling refers to, we will write [x] u .

The set C ∇u
We recall here some results from [5,13] that will be essential in later sections. In what follows, if U is a connected component of C ∇u , then for each x ∈ S ∩ ∂U the segment [x] is called an edge of U. To prove the nal assertion, notice that if [x] is not contained in ∂U, then it must be disjoint from all edges of U. Since, however, [x] intersects U, the segment [x] must have precisely one endpoint in common with an edge l of U. Hence, by part (b), [x] and l belong to the boundary of a connected component V of C ∇u . Since V and U have zero distance from each other (both contain l in their boundary), the continuity of ∇u and the maximality of U imply that It is useful to notice that if u is a minimizer in A u 0 (S, ∂ c S), then it is also a minimizer under its own boundary conditions on suitable subsets of S. Proposition 2.11. Let S be a bounded simply connected domain, let l 1 , . . . , l k ⊂ S be pairwise disjoint line segments such that there is a connected component Then for every minimizer u within A u (S, ∂ c S) such that each l i (with i = 1, . . . , k) is a segment of constancy of ∇u, the restriction u| S 0 is a minimizer within

Proof. The result follows from the observation that ifũ
belongs to the set A u (S, ∂ c S). Details can be found in ( [5], proposition 2.11).

Remark. Notice that
iso (S; R 3 ), let l 1 , . . . , l k ⊂ S be pairwise disjoint segments of constancy of ∇u, and let S 0 be a connected component of S\ k i=1 l i . Assume that u is a minimizer within Then the following are true:

Negative regularity result
The Euler-Lagrange equations derived in [12] lead to the (positive) regularity result theorem 1.1. However, they can also be used to obtain a negative regularity result which will be essential in the examples below. It is based on the following lemma.
. Denote by ζ the inner unit normal to B + R in x 0 and denote by ∂ ζ the directional derivative in the direction ζ. Then u ∈ C 3 (B + R ) and one of the following mutually exclusive assertions is true: The proof of lemma 2.13 depends heavily upon [12,13] and is therefore postponed to the appendix.
In what follows, for a given α ∈ (0, 1) and an open set V ⊂ R 2 we denote by C 3,α (V ) the space of all u ∈ C 3 (V ) which satisfy the Hölder condition Let us note some consequences of lemma 2.13.
Lemma 2.14. Let ∂ c S, u 0 and u satisfy the hypotheses of theorem 1.1. Let α ∈ (0, 1), let r > 0 and let x 0 ∈ S be such that B r (x 0 ) ⊂ S and assume that u ∈ C 3,α (B r (x 0 )). Assume, moreover, that u is af ne on one half-ball (of B r (x 0 )) and that the other half-ball does not intersect Σ c ∪ Σ τ ∪ ∂ C ∇u . Then u is af ne on all of B r (x 0 ).

Proof.
We assume without loss of generality that x 0 = 0 and that u is af ne on B − r = {x ∈ B r (0) : x · e 1 < 0}. Since u ∈ C 1 (B r (0)), it is af ne along B r (0) ∩ {0} × R. Hence the hypotheses of lemma 2.13 are satis ed.
Notice that all derivatives of u of order at least two vanish on B − r . Since u ∈ C 2 (B r (0)), we therefore have |∂ 1 ∂ 1 u(x)| → 0 as x → 0. Hence (a) from lemma 2.13 does not hold.
Moreover, since u ∈ C 3,α (B r (0)), for all t ∈ (0, r) we have Hence neither (b) nor (c) of lemma 2.13 are satis ed. Hence statement (d) from that lemma must be true.

Proposition 2.15.
Let ∂ c S, u 0 and u satisfy the hypotheses of theorem 1.1 and let α ∈ (0, 1). Let U be a connected component of C ∇u , let x 0 ∈ S ∩ ∂U and denote the outer unit normal to U in x 0 by ζ. Assume that there is an R > 0 such that Then u fails to be C 3,α in any neighbourhood of x 0 .
Proof. Due to lemma 2.9 there is r ∈ (0, R) such that one connected component of B r (x 0 )\[x 0 ] is contained in U, while the other one, namely the half-disk does not intersect U. So if we had u ∈ C 3,α (B r (x 0 )) for some α ∈ (0, 1), then B r (x 0 ) would satisfy the hypotheses of lemma 2.14. Hence u would have to be af ne on all of B r (x 0 ), so by maximality of U the whole disk B r (x 0 ) would have to be contained in U, contradicting the properties of B + r .

Corollary 2.16.
Let ∂ c S, u 0 and u satisfy the hypotheses of theorem 1.1 and let α ∈ (0, 1). Let U be a connected component of C ∇u and let x 0 ∈ (S ∩ ∂U)\(Σ τ ∪ Σ c ). Assume, moreover, that x 0 has a positive distance from C ∇u \U (this is satis ed, e.g. if C ∇u has only nitely many connected components). Then u fails to be C 3,α near x 0 .
Proof. We use the notation from the proof of proposition 2.15. Since x 0 has a positive distance from C ∇u \U, after possibly shrinking r we may assume that B + r does not intersect C ∇u .
Since S\(Σ c ∪ Σ τ ) is open and contains x 0 , we may further assume that B + r is contained in this set as well. Hence the claim follows from proposition 2.15.

Counterexamples to regularity
We are now ready to construct the examples announced in section 1.2.

Cylindrical building block
For the examples given in the next sections, it is useful to have the following building block at hand: with any arclength parametrised curve Ξ from some interval I into R 2 and with any ξ ∈ S 1 we associate the corresponding cylindrical deformation de ned on J Ξ,ξ = {x ∈ R 2 : x · ξ ∈ I}, with values in R 3 and using the standard identi cation is a local parametrisation of the cylinder generated by the regular curve Ξ. For the particular case Ξ(t) = (sint, 1 − cost), we will brie y write Z ξ instead of Z Ξ,ξ .
Observe that Z Ξ,ξ is an isometric immersion. Indeed, one computes from which directly follows that Z Ξ,ξ is an isometric immersion on its domain; its regularity is clearly determined by that of the curve Ξ.

Example concerning Σ c
Here we give a simple example of a smooth convex domain S and boundary data u 0 ∈ W 2,2 iso (S; R 3 ) and ∂ c S such that the minimizer is not C 2 and for which we can fully characterise . Using the cylindrical deformation from section 3.1 with the vector ξ = e 1 , we de ne where we identify R 2 with the set R 2 × {0} ⊂ R 3 . We can write this explicitly as u 0 (x) = (sin(x 1 ), x 2 , 1 − cos(x 1 )) T for x 1 > 0, (x 1 , x 2 , 0) T for x 1 0.
The situation is depicted in gure 2. Proposition 3.1. Let u ∈ A u 0 (S, ∂ c S). Then the following are true: whenever (x 1 , x 2 ) ∈ ∂ c S. Hence (a) follows from remark 2.2.
To prove (c), notice that by (a) we know that u is af ne on (S ∩ R)\B + 1 (0). Since (S\R) ∩ ∂ c S = ∅ and u is a minimizer, it follows that u is also af ne on S\R. Hence

Example concerning ∂ C ∇u
In this section we give an example of a domain and of boundary data which show that the set C ∇u can really be nonempty for a minimizer (and indeed for any admissible immersion u ∈ A u 0 (S, ∂ c S)). We then show that regularity may indeed fail at the singular set S ∩ ∂ C ∇u .
3.3.1. Existence of C ∇u . We give an example of a smooth convex domain S and boundary data u 0 ∈ W 2,2 iso (S; R 3 ) and ∂ c S such that C ∇u = ∅ for all u ∈ A u 0 (S, ∂ c S). Let S = B 1 (0) ⊂ R 2 and let ∆ denote the interior of the equilateral triangle with baricenter at the origin and vertices on ∂S given by e −i π 6 , e i π 2 and e i 7π 6 ; see gure 3.
Here and in what follows we use the standard identi cation of R 2 with C. Setting ϕ k = π 6 + 2π 3 k and v k = e iϕ k , we have The idea is to choose u 0 ∈ W 2,2 iso (S; R 3 ) such that u 0 | ∆ is the identity, while on the three connected components of S\∆ the deformation u 0 consists of pieces of cylinders that are C 1attached to ∆. By suitably choosing ∂ c S ⊂ ∂S, any element u in A u 0 (S, ∂ c S) will necessarily satisfy C ∇u = ∅.
We de ne u 0 : S → R 3 explicitly by setting (we use the canonical injection of R 2 into R 3 ) Here Z v k the cylindrical deformation de ned in section 3.1. The map u 0 is perhaps the simplest isometric deformation displaying the set C ∇u ; as such it resembles, e.g. a gure in [23], where the set C ∇u is discussed in the context of rigidity. Notice that one could easily modify u 0 to make the transition from the af ne triangle to each cylindrical part (C ∞ -)smooth. For minimizers, however, we will see that this transition is not smooth. Our rst main result in this section is the following one: Let v k and u 0 as de ned above, let α ∈ ( π 3 , 2π 3 ) and set ρ = 2 cos α 2 .

De ne
Then for every u ∈ A u 0 (S, ∂ c S) the set C ∇u is nonempty and contains the origin.
The upper bound on α ensures that ρ > 1 while the lower bound is needed in the proof of lemma 3.3 below.
We split the proof of proposition 3.2 into two lemmas. Set x ∈ R 2 : x · v k < ρ 2 and de ne S ′ = S ∩ ∆ ′ . For k = 0, 1, 2 set l k = {x ∈ S : x · v k = ρ/2}. We rst prove that any segment with both endpoints on ∂S and containing the origin has to intersect S\S ′ . Lemma 3.3. Let l ⊂ S be a segment containing the origin and intersecting ∂S at both endpoints. Then there is k ∈ {0, 1, 2} and there are y (1) , y (2) ∈ l such that Proof. First we notice that the claim will follow directly once we prove that l intersects S\S ′ , because S\S ′ is open. We parameterised l by the map (−1, 1) ∋ t → te iψ for some ψ ∈ R. By the choice of ρ we see that the ray R + e iψ intersects S\S ′ if and only if for some k. So if l ⊂ S is an arbitrary segment through the origin with end-points e iψ , e i(ψ+π) on ∂S, then l intersects S\S ′ if and only if ψ is contained in The equality arises from the de nition of the ϕ k . But the right-hand side is the whole real line because α > π 3 . Hence the condition on ψ is trivially satis ed. Lemma 3.4. Let S ′ be de ned as in (11). Then, u = u 0 on S\S ′ for any u ∈ A u 0 (S, ∂ c S).
Proof. Clearly, C ∇u 0 = ∆ and the ruling q u 0 of ∇u 0 is such that q u 0 (x) = v ⊥ k whenever x · v k 1/2. The claim follows from lemma 2.8.
Proof of proposition 3.2. Let l be a segment in S with both endpoints on ∂S and containing the origin. Let y (1) , y (2) ∈ l as in the conclusion of lemma 3.3. From the de nition of u 0 and from (9) it is clear that ∇u 0 ( y (1) ) = ∇u 0 ( y (2) ) since y (1) · v k = y (2) · v k . On the other hand, by lemma 3.4 we also have u = u 0 on the open set S\S ′ , to which y (1) and y (2) belong. So ∇u( y (1) ) = ∇u( y (2) ). We conclude that ∇u is not constant on l.
Hence there exists no segment of constancy of ∇u that contains the origin. So by remark 2.6 the origin is contained in C ∇u .
3.3.2. Failure of regularity for minimizers. We will now see that minimizers may indeed fail to be C ∞ at ∂ C ∇u . With notation as above, we have the following result.
We begin with a general lemma.
Lemma 3.5. Let S be convex, ∂ c S ⊂ ∂S closed and let u ∈ W 2,2 iso (S). Then Remark. For nonconvex S the assertion is false in general.

Proof.
We must show that the set on the right-hand side of (14) is closed in S. Let x n ∈ S\C ∇u be such that [x n ] ∩ ∂ c S = ∅ for all n and assume that the x n converge to some x ∈ S. Then x / ∈ C ∇u because C ∇u is open. And by ( [13], lemma 2(iii)) the segments [x n ] converge to [x] in the Hausdorff sense. Hence indeed [x] ∩ ∂ c S = ∅ as well. Proposition 3.6. Let S = B 1 (0), de ne ∂ c S as in (13) and u 0 as in (12), and let u be a minimizer within A u 0 (S, ∂ c S).
Then C ∇u consists of precisely one connected component U. The component U contains the origin and it has precisely three edges. At least two of these edges do not belong to Σ c . And near any point in S ∩ ∂U\Σ c (i.e. on any edge that does not belong to Σ c ) the map u fails to be C 3,α for any α > 0.
Proof. As u ≡ u 0 on S\S ′ , by lemma 3.4 it is clear that C ∇u ⊂ S ′ , because ∇u 0 is not constant on any open subset of S\S ′ . By proposition 3.2 we know that there exists a connected component U of C ∇u containing the origin, and that in fact U belongs to C ∇u .
Clearly S ′ is a connected component of S\ 2 k=0 l k and S ∩ ∂S ′ = 2 k=0 l k . By continuity ∇u is constant on each l k and agrees with ∇u 0 on l k , because both agree on S\S ′ . Notice that l k = [x k ] for suitable x k ∈ S\C ∇u , because clearly l k is not contained in C ∇u (since C ∇u ⊂ S ′ ), and ∇u is constant on l k . In particular, the segments l k are disjoint segments of constancy of ∇u.

Proposition 2.11 implies that u| S ′ is a minimizer within
Since C ∇u ⊂ S ′ , in particular U ⊂ S ′ . Proposition 2.12 implies that U is the only connected component of C ∇u and that it has at most (hence precisely, due to the de nition of C ∇u ) three edges. Lemma 2.10(b) shows that for k = 0, 1, 2 we have Since ∇u is constant on U and since ∇u = ∇u 0 attains a different constant value on each of the l k , we deduce from (16) that there is at most one k such that U intersects l k (and then in fact l k ⊂ ∂U ). Hence lemma 3.5 implies that the other two edges of U (they constitute the set S ∩ ∂U\l k ) do not intersect Σ c . Finally, let x 0 ∈ S ∩ ∂U\Σ c . Since S is convex we have Σ τ = ∅. Since moreover C ∇u = U, we conclude that x 0 satis es the hypotheses of corollary 2.16, so the claim follows.

Example concerning
and de ne Set and ∂ + c S = {x ∈ ∂S : x 1 < 0 and x 2 > Lemma 3.7. There exists an isometric immersion u 0 ∈ W 2,2 iso (S) that satis es the boundary conditions Intuitively, any u ∈ A u 0 (S; ∂ c S) therefore remains clamped to the plane {x 3 = 0} on the lower part of S, while the left 'hill' of S is bent backwards by an angle π.
Proof. The deformation u 0 will be cylindrical. The underlying two-dimensional curve is β : R → R 2 given by Notice that β ∈ W 2,∞ (R) and that it is parametrised by arclength. We de ne Ξ : R → R 2 by This curve is depicted in gure 4. Finally, we set u 0 = Z Ξ,e 2 ; the right-hand side is the cylindrical building block from section 3.1. It is easy to verify that u 0 has the desired properties.
As was the case in our previous example, also here we could easily modify u 0 to become a C ∞ isometric immersion satisfying (19). This merely requires replacing the curve β by a smooth curve with similar properties, e.g. by mollifying and reparametrising.
We will see that a typical u ∈ A u 0 (S, ∂ c S) must exhibit the singular set Σ τ . Before stating this result, we collect some simple facts about f.  (20) and

Moreover, we have
Proof. Most statements are readily veri ed. Regarding inequality (20), it follows from the fact that 7+ √ 7 7 < 12 7 . Finally, inequality (21) follows from Proposition 3.9. Let S be as in (17), let ∂ c S as in (18), let u 0 satisfy the conclusion of lemma 3.7 and assume that u is a minimizer within A u 0 (S, ∂ c S). Then there exists an x 0 ∈ S\C ∇u such that [x 0 ] intersects ∂S tangentially. Moreover, u fails to be C 3,α near x 0 , for any α ∈ (0, 1).
Proof. Since ∂ c S consists of only two disjoint components ∂ − c S and ∂ + c S, an application of ( [5], proposition 4.9) shows that C ∇u is empty. Hence by ( [13], proposition 1) and ( [13], proposition 9), there exists a ruling for u on S\ C ∇u , which in this case is all of S. We x one of these (not uniquely determined) rulings on S and call it q. As usual, we write Let a = 1 √ 7 and de ne the interval Set z 0 = sup Z. Observe that Z is nonempty and z 0 > 0 because q(a, z) → ±e 1 as z ↓ 0. We claim that z 0 ∈ (0, f(a)) and that [(a, z 0 )] intersects graph f| (−a,a) tangentially, see gure 5. Denote by m(z) the unique slope such that the segment [(a, z)] lies on the graph of the af ne function l z : R → R de ned by Let us prove that for all z ∈ (0, f(a)). In fact, otherwise l z < f on [a, 1] and l z (1) < 0. The former inequality implies that the graph of l z restricted to {x 1 a : l z (x 1 ) > 0} agrees with [(a, z)] ∩ {x 1 a}. Hence the inequality l z (1) < 0 would imply that [(a, z)] intersects R × {0}. This would contradict the properties of the ruling q.
Next we claim that In fact, assume that this were not satis ed for some z ∈ Z. Then l z (−1) < 0. On the other hand, by the de nition of Z we know that [(a, z)] does not intersect the graph of f| (−a,a) . Hence l z < f on [−1, a]. We can argue as before to see that this would imply that [(a, z)] intersects R × {0}.
For z ∈ Z we have l z (0) < f(0), because otherwise l z would intersect the graph of f in [0, a). Hence we conclude using (23) that Since f(a) exceeds the right-hand side of this inequality by (20), we conclude that indeed z 0 < f(a). We claim that l z 0 intersects f tangentially in precisely one point a 1 ∈ [−a, a], and that in fact a 1 ∈ (−a, a). To see this, observe that by maximality of z 0 and compactness of graphf [−a,a] , it clearly must intersect f in [−a, a]. And then it must do so at some point a 1 ∈ (−a, a) because l z 0 (a) = z 0 < f (a), so l z 0 cannot intersect f in a. But if it intersected f in −a, then its slope m(z 0 ) would have to be smaller than f ′ (−a). Therefore using (21) and (24), we would derive contradicting (22). Finally, let us verify that l z 0 must intersect f tangentially in a 1 . In fact, otherwise for all z near z 0 the function l z would intersect f on (−a, a) as well, because a 1 ∈ (−a, a) and l z → l z 0 locally uniformly on R as z → z 0 . This would contradict the choice of z 0 .
By strict convexity of f| (−a,a) , we see that l z 0 intersects f precisely once on [−a, a], because it is a supporting segment.
We have seen that l z 0 intersects f on [−a, a] in precisely one point a 1 ∈ (−a, a). Since f is strictly concave outside [−a, a], there is exactly one a 2 ∈ (a, 1] with l z 0 (a 2 ) = f (a 2 ). Clearly [(a, z 0 )] = graph l z 0 | (a 1 ,a 2 ) . The set U = {(x 1 , x 2 ) : x 1 ∈ (a 1 , a 2 ) and x 2 ∈ (l z 0 (x 1 ), f (x 1 ))} is a connected component of S\[(a, z 0 )] that has a positive distance from ∂ c S. Since u is a minimizer, it is af ne on U. Therefore, in view of (25), we conclude that x 0 = (a, z 0 ) satis es the hypotheses of proposition 2.15. The claim follows from that proposition.
Finally, notice that F(τ ) cF(x 1 ) for some c > 0 because of (39). The remaining estimate is proven similarly.