Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine away from a closed, countably $\mathcal{H}^{n - 1}$-rectifiable set. If $A$ is affinely independent, then $u$ is piecewise affine away from a closed $\mathcal{H}^{n - 1}$-null set.


Introduction
For n ∈ N, consider a finite set A ⊆ R n . We study continuous functions u : R n → R such that the weak gradient ∇u satisfies ∇u ∈ BV loc (R n ; R n ) and ∇u(x) ∈ A for almost every x ∈ R n . This means that whenever Ω ⊆ R n is open and bounded, the sets {x ∈ Ω : ∇u(x) = a}, for a ∈ A, form a Caccioppoli partition of Ω as discussed, e.g., by Ambrosio, Fusco, and Pallara [1, Section 4.4]. The theory of Caccioppoli partitions therefore applies and gives some information on the structure of ∇u and of u. The fact that we are dealing with a gradient, however, gives rise to a better theory, especially under additional assumptions on the geometry of A. We work with the following notions in this paper.
Definition 1. A set A ⊂ R n is called convex independent if any a ∈ A does not belong to the convex hull of A \ {a}. It is called affinely independent if any a ∈ A does not belong to the affine span of A \ {a}.
If either of these conditions is satisfied, then we can prove statements on the regularity of u that finite Caccioppoli partitions do not share in general. In fact, we will see that u is locally piecewise affine away from a closed, countably H n−1rectifiable set (if A is convex independent) or away from a closed H n−1 -null set (if A is affinely independent).
In order to make this more precise, we introduce some notation. Given r > 0 and x ∈ R n , we write B r (x) for the open ball of radius r centred at x. Given a ∈ R n , the function λ a : R n → R is defined by λ a (x) = a · x for x ∈ R n . Given two functions v, w : R n → R, we write v ∧ w and v ∨ w, respectively, for the functions with (v ∧ w)(x) = min{v(x), w(x)} and (v ∨ w)(x) = max{v(x), w(x)} for x ∈ R n . Definition 2. Given a function u : R n → R, the regular set of u, denoted by R(u), consists of all x ∈ R n such that there exist a, b ∈ R n , c ∈ R, and r > 0 with u = λ a ∧ λ b + c in B r (x) or u = λ a ∨ λ b + c in B r (x). The singular set of u is its complement S(u) = R n \ R(u).
The condition for R(u) allows the possibility that a = b, in which case u is affine near x. If a = b, then it is still piecewise affine near x. Obviously R(u) is an open set and S(u) is closed.
It would be reasonable to include functions consisting of more than two affine pieces in the definition of R(u), for example (λ a1 ∧ λ a2 ) ∨ λ a3 + c for a 1 , a 2 , a 3 ∈ R n and c ∈ R. For the results of this paper, however, this would make no difference, therefore we choose the simpler definition.
For s ≥ 0, we denote the s-dimensional Hausdorff measure in R n by H s . The notation BV 2 loc (R n ) is used for the space of functions with weak gradient in BV loc (R n ; R n ). Thus the hypotheses of the following theorems are identical to the assumptions at the beginning of the introduction.
Theorem 3. Suppose that A is a finite, convex independent set. Let u ∈ BV 2 loc (R n ) with ∇u(x) ∈ A for almost every x ∈ R n . Then S(u) is countably H n−1 -rectifiable.
Theorem 4. Suppose that A is a finite, affinely independent set. Let u ∈ BV 2 loc (R n ) with ∇u(x) ∈ A for almost every x ∈ R n . Then H n−1 (S(u)) = 0.
For n = 2, Theorem 4 was proved in a previous paper [10]. For higher dimensions, the result is new. Theorem 3 is new even for n = 2. For n = 1, both statements are easy to prove.
The results are optimal in terms of the Hausdorff measures involved. Furthermore, the assumption of convex/affine independence is necessary. Indeed, there are examples of finite sets A ⊆ R 2 and functions u ∈ BV 2 loc (R 2 ) with ∇u(x) ∈ A almost everywhere such that • H 2 (S(u)) > 0; or • H 1 (S(u)) > 0 and A is convex independent; or • H s (S(u)) = ∞ for any s < 1 and A is affinely independent.
All of these can be found in the author's previous paper [10].
Apart from being of obvious geometric interest, functions as described above appear in problems from materials science. They naturally arise as limits in Γconvergence theories in the spirit of Modica and Mortola [8,9] for quantities such asˆΩ where Ω ⊆ R n is an open set and W : R n → [0, ∞) is a function with A = W −1 ({0}). Functionals of this sort appear in certain models for the surface energy of nanocrystals [13,7,14]. For Ω ⊆ R 2 , functions u ∈ BV 2 (Ω) with ∇u ∈ {(±1, 0), (0, ±1)} have also been used by Cicalese, Forster, and Orlando [3] for a different sort of Γ-limit arising from a model for frustrated spin systems. Functionals similar to (1), but for maps u : Ω → R n , also appear in certain models for phase transitions in elastic materials (see, e.g., the seminal paper of Ball and James [2] or the introduction into the theory by Müller [11]). In this context, due to the frame indifference of the underlying models, the set W −1 ({0}) is typically not finite. Sometimes, however, the frame indifference is disregarded (as in the paper by Conti, Fonseca, and Leoni [4]), or the theory gives a limit with ∇u ∈ BV(Ω; A) for a finite set A ⊆ R n×n anyway (such as in recent results of Davoli and Friedrich [6,5]). In such a case, Theorem 3 and Theorem 4 are potentially useful, as they apply to the components (or other one-dimensional projections) of u.
In the proof of Theorem 4, we use some of the tools from the author's previous paper [10]. In particular, we will analyse the intersections of the graph of u with certain hyperplanes in R n+1 . We will see that these intersections correspond to the graphs of functions with (n − 1)-dimensional domains and with properties similar to u. The key ideas from the previous paper, however, are specific to R 2 , so we eventually use different arguments. In this paper, we use the theory of BV loc (R n ; R n ) to a much greater extent. The central argument will consider approximate jump points of ∇u. Near such a point, we know that u is close to a piecewise affine function in a measure theoretic sense by definition. We then use an induction argument (with induction over n) to show that u is in fact piecewise affine near H n−1 -almost every approximate jump point.
We also need to analyse points where u has an approximate limit, and they are of interest for the proofs of both Theorem 4 and Theorem 3. This part of the analysis is significantly simpler and relies on the fact that for any a ∈ A, the function v(x) = u(x) − a · x has some monotonicity properties.
In the rest of the paper, we study a fixed function u ∈ BV 2 loc (R n ) with ∇u(x) ∈ A for almost every x ∈ R n . Since we are interested only in the local properties of u, we may assume that it is also bounded. (Otherwise we can modify it outside of a bounded set with the construction described in [10,Section 6].) We define the function U : R n → R n+1 by , x ∈ R n .
We use the notation graph(u) = U (R n ) for the graph of u.
As we sometimes work with points in R n+1 (especially points on graph(u)) and their projections onto R n simultaneously, we use the following notation. A generic point in R n+1 is denoted by x = (x 1 , . . . , x n+1 ) T , and then we write x = (x 1 , . . . , x n ) T . Thus x = ( x xn+1 ). We think of elements of R n and of R n+1 as column vectors, and this is sometimes important, as we use them as columns in certain matrices.
As our function satisfies in particular the condition ∇u ∈ BV loc (R n ; R n ), the theory of this space will of course be helpful. In this context, we mostly follow the notation and terminology of Ambrosio, Fusco, and Pallara [1]. We also use several of the results found in this book.

Approximate faces and edges of the graph
In this section, we decompose R n into three sets F , E, and N . These are defined such that we expect regularity in F under the assumptions of either of the main theorems, and also in E under the assumptions of Theorem 4. The third set, N , will be an H n−1 -null set. The sets F and E characterised, up to H n−1 -null sets, by the condition that ∇u has an approximate limit or an approximate jump, respectively. Since much of our analysis examines graph(u), it is also convenient to think of F as the set of points where the graph behaves approximately like the (n-dimensional) faces of a polyhedral surface, whereas E corresponds to approximate ((n − 1)-dimensional) edges.
First we define the set F ′ ⊆ R n , comprising all points x ∈ R n such that there exists a ∈ R n satisfying lim rց0 Br (x) |∇u − a| dH n = 0.
In other words, this is the set of all points where ∇u has an approximate limit a. It is then clear that a ∈ A. The complement R n \ F is called the approximate discontinuity set of ∇u.
Furthermore, let E ′ be the set of all x ∈ R n such that there exist a − , a + ∈ R n with a − = a + and there exists η ∈ S n−1 such that and lim This is the approximate jump set of ∇u. Again, the points a − , a + will always belong to A. According to a result by Federer and Vol'pert (which can be found in the book by Ambrosio, Fusco, and Pallara [1, Theorem 3.78]), there exists an H n−1null set N ′ ⊆ R n such that Furthermore, the set E ′ is countably H n−1 -rectifiable. Given x ∈ R n and ρ > 0, we define the function u x,ρ : R n → R with forx ∈ R n . For x fixed, the family of functions (u x,ρ ) ρ>0 is clearly bounded in C 0,1 (K) for any compact set K ⊆ R n . Therefore, the theorem of Arzelà-Ascoli implies that there exists a sequence ρ k ց 0 such that u x,ρ k converges locally uniformly. If we have in fact a limit for ρ ց 0, then we write and call this limit the tangent function of u at x. If x ∈ F ′ and a ∈ A is the approximate limit of ∇u at x, then for any sequence ρ k ց 0, the limit of u x,ρ k can only be λ a . Hence in this case, there exists a tangent function T x u, which is exactly this function. Similarly, if x ∈ E ′ , then T x u exists and Because T x u is a continuous function, this means that Then we conclude that T x u = λ a− ∧ λ a+ or T x u = λ a− ∨ λ a+ , depending on the sign. If we consider the functions a − , a + : E ′ → A and η : E ′ → S n−1 such that (2) and (3) are satisfied on E ′ , then the previously used result [1, Theorem 3.78] also implies that Let γ = min {|a − b| : a, b ∈ A}. Then for any Borel set Ω ⊆ R n , we conclude that |D∇u|(Ω) ≥ γH n−1 (E ′ ∩ Ω).

Now define
Then standard results [1, Theorem 2.56 and Lemma 3.76] imply that H n−1 (F ′ \ F ) = 0. Recall the map U : R n → R n+1 defined in the introduction. Set F * = U (F ) and E † = U (E ′ ). Then E † is a countably H n−1 -rectifiable subset of R n+1 . Hence at H n−1 -almost every x ∈ E † , the measure H n−1 E † has a tangent measure [1, Theorem 2.83] of the form H n−1 T x E † , where T x E † is an (n − 1)-dimensional linear subspace of R n+1 (the approximate tangent space of E † at x). Let E * be the set of all x ∈ E † where this is the case. Furthermore, let E = U −1 (E * ). Then E ′ \ E is an H n−1 -null set.
Thus if we define N = R n \ (F ∪ E), then N is an H n−1 -null set and we have the disjoint decomposition

Proof of Theorem 3
In this section we prove our first main result, Theorem 3. The proof is based on the following proposition, which will also be useful for the proof of Theorem 4 later on.
Proposition 5. Suppose that A ⊆ R n is finite and convex independent. Let u ∈ BV 2 loc (R n ) be a function with ∇u(x) ∈ A for almost all x ∈ R n . Then there exist r > 0 and ǫ > 0 with the following property. Suppose that there exists and |D∇u|(B 1 (0)) ≤ ǫ.
Then ∇u(x) = a for almost every x ∈ B r (0).
Proof. Because A is convex independent, there exists ω ∈ S n−1 such that As A is finite, there also exists δ ∈ (0, 1) such that the inequality a · ξ ≤ min b∈A\{a} (b · ξ) holds even for ξ in the cone Thus v is monotone along lines parallel to ξ. (This is true for every such line by the continuity of v.) Furthermore, for almost every x ∈ R n , we find that either ∇u( We now foliate a part of B 1 (0) by line segments parallel to ω.
Provided that r is chosen sufficiently small, we can find R ∈ (0, 1] such that In particular, the restriction of v to the line segment L z is not constant. For On the other hand, because of (4), we also know that Set c = min b∈A |a − b|. For H n−1 -almost any z ∈ Z ′ , the function t → ∇u(z + tω) belongs to BV (− 1 2 , 1 2 ); R n and its total variation is at least c.
If ǫ is sufficiently small, then this means in particular that |D∇u|(B 1 (0)) > ǫ. Thus we have proved the contrapositive of Proposition 5.
Proof of Theorem 3. We show that F ⊆ R(u). To this end, fix x ∈ F and consider the rescaled functions u x,ρ for ρ > 0. Since x ∈ F , we know that ∇u x,ρ → a in L 1 (B 1 (0)) as ρ ց 0 for some a ∈ A. Furthermore, since as ρ ց 0, the function u x,ρ satisfies the inequalities of Proposition 5 for ρ sufficiently small. Hence ∇u x,ρ (x) = a for almost everyx ∈ B r (0), which implies that for allx ∈ B ρr (x). Hence x ∈ R(u). Theorem 3 now follows from the observations in Section 2.

Specialising to a regular n-simplex
The rest of the paper is devoted to the proof of Theorem 4. Instead of considering any affinely independent set A, we now assume that a 0 , . . . , a n ∈ R n are the corners of a regular n-simplex of side length √ 2n + 2 centred at 0, and that A = {a 0 , . . . , a n }. We further assume that the matrix with columns a 0 − a 1 , . . . , a 0 − a n has a positive determinant. Theorem 4 can then be reduced to this situation by composing u with an affine transformation. The details are given on page 24 below.
As it is sometimes convenient to permute a 0 , . . . , a n cyclically, we regard 0, . . . , n as members of Z n+1 = Z/(n + 1)Z in this context. Thus a i+n+1 = a i .
The condition that our simplex has side length √ 2n + 2 means that |a i | = √ n for every i ∈ Z n+1 . Indeed, by the calculations of Parks and Wills [12], the dihedral angle of the regular n-simplex is arccos 1 n . As each a i is orthogonal to one of the faces, this means that a i · a j = − 1 n |a i ||a j | for i = j, and therefore 2n + 2 = |a i − a j | 2 = 2n+2 n |a i ||a j |. From this we conclude that Then Hence (ν 1 , . . . , ν n+1 ) is an orthonormal basis of R n+1 . (This is the reason why we choose A as above.) Furthermore, det −a 1 · · · −a n+1 1 · · · 1 = det a 0 − a 1 · · · a 0 − a n −a 0 0 · · · 0 1 = det a 0 − a 1 · · · a 0 − a n .
(In the first step, we have used the fact that a n+1 = a 0 and subtracted the last column from each of the other columns of the matrix.) Hence the above assumption guarantees that the basis (ν 1 , . . . , ν n+1 ) gives the standard orientation of R n+1 . We now use the notation λ i = λ ai , recalling that this is the linear function Thus we have the disjoint decomposition Thus in order to understand F , E, or F i , it suffices to study F * , E * , or F * i and how U −1 transforms them. In particular, the following is true.
Proof. We use the area formula [1, Theorem 2.91]. Hence we need to calculate the Jacobian of U restricted to the approximate tangent spaces of E. More precisely, since E is countably H n−1 -rectifiable, there exists an approximate tangent space T x E at H n−1 -almost every x ∈ E. Because U is Lipschitz continuous, the tangential derivative d E U (x) exists at H n−1 -almost every x ∈ E [1, Theorem 2.90]. We write L * for the adjoint of a linear operator L. Then is the Jacobian of U at x with respect to T x E. The area formula implies that Thus in order to prove the first identity, it suffices to show that as ρ ց 0. The convergence is in fact uniform on compact subsets of (a i − a j ) ⊥ .

Slicing the graph
We still assume that A consists of the corners of the regular n-simplex from Section 4 and we assume that u ∈ BV 2 loc (R n ) is bounded and satisfies ∇u(x) ∈ A for almost every x ∈ R n . In this section, we analyse the graph of u. In particular, we examine intersections of graph(u) with hyperplanes perpendicular to one of the vectors ν i . We will see that almost all such intersections can be represented as the graphs of functions in BV 2 loc (P ), where P = {y ∈ R n : y 1 + · · · + y n = 0} , and with gradient taking one of n different values almost everywhere. That is, we have a function with properties similar to u, but with an (n − 1)-dimensional domain. This observation will eventually make it possible to prove Theorem 4 with the help of an induction argument. We use some tools from the author's previous paper [10] in this section. Given i ∈ Z n+1 , let Φ i : R n+1 → R n+1 be the linear map with This corresponds to the intersection of graph(u) with a hyperplane orthogonal to ν i after rotation by Φ i , or in other words, a slice of graph(u). We further define the functions g i (y) = sup t ∈ R : u(tν i + y 1 ν i+1 + · · · + y n ν i+n ) > t + y 1 + · · · + y n √ n + 1 and g i (y) = inf t ∈ R : u(tν i + y 1 ν i+1 + · · · + y n ν i+n ) < t + y 1 + · · · + y n √ n + 1 .
Note that for a fixed y ∈ R n , the set t ∈ R : u(tν i + y 1 ν i+1 + · · · + y n ν i+n ) = t + y 1 + · · · + y n √ n + 1 corresponds to the intersection of graph(u) with a line parallel to ν i , so the functions g i and g i tell us something about the geometry of graph(u) as well.
The following properties of g i and g i have been proved elsewhere for n = 2 [10, Lemma 16]. The proof carries over to higher dimensions as well. We therefore do not repeat it here.
Lemma 7. For any i ∈ Z n+1 , the following statements hold true.
(i) The function g i is lower semicontinuous and g i is upper semicontinuous.
(ii) The identity g i = g i holds almost everywhere in R n .
(iii) For any y ∈ R n , the inequality g i (y) ≤ g i (y) holds true and (iv) Let t ∈ R and y ∈ R n . Then y ∈ Γ i (t) if, and only if, g i (y) ≤ t ≤ g i (y).
(vi) For all y ∈ R n and all ζ ∈ [0, ∞) n , the inequalities g i (y) ≥ g i (y + ζ) and g i (y) ≥ g i (y + ζ) are satisfied.
Now consider the hyperplane P ⊆ R n given by P = {y ∈ R n : y 1 + · · · + y n = 0} and its unit normal vector for i = j. Hence b 1 , . . . , b n are the corners of a regular (n − 1)-simplex in P centred at 0 with side length √ 2n. (Indeed the construction is similar to the standard (n − 1)-simplex.) Thus they are the (n − 1)-dimensional counterparts to a 0 , . . . , a n .
Given a function f : P × R → R, we write∇f for its gradient with respect to the variable p ∈ P . We want to show the following.
Proposition 8. Let i ∈ Z n+1 . Then there exists a function f i : P × R → R such that for almost every t ∈ R, • the function p → f i (p, t) belongs to BV 2 loc (P ) and∇f i (p, t) ∈ {b 1 , . . . , b n } for H n−1 -almost every p ∈ P ; and Before we can prove this result, we need a few lemmas.
Define ζ − = y − y − and ζ + = y + − y. Then ζ − , ζ + ∈ (0, ∞) n . According to Lemma 7, this means that Hence y ∈ Γ i (t). By the semicontinuity of g i and g i , we also conclude that Lemma 10. Let i ∈ Z n+1 . Let t ∈ R and p ∈ P . Suppose that Then Proof. Let y ∈ p + s − σ − (0, ∞) n . Choose s < s − such that y ∈ p + sσ − (0, ∞) n as well. Then Lemma 7 implies that Hence y ∈ Γ i (t). The proof of the second statement is similar.
Lemma 11. There exists a constant C such that the following holds true. Suppose that v : R n → R is smooth and bounded with a j · ∇v > −1 for all j ∈ Z n+1 and sup R n |v| ≤ M . Let i ∈ Z n+1 . Let φ : P × R → R be the unique function such that for p ∈ P and t ∈ R. Then |∇φ(p, t)| ≤ √ n for all p ∈ P and t ∈ R. Moreover, for any R > 0, Since the proof of this statement is lengthy, we postpone it to the next section. We now prove Proposition 8.
Proof of Proposition 8. Let t ∈ R and p ∈ P . Since u is bounded, the line If there are s − , s + ∈ R with s − < s + such that p + s − σ ∈ Γ i (t) and p + s + σ ∈ Γ i (t), then Lemma 9 implies that Γ i (t) has non-empty interior, denoted byΓ i (t). Because of Lemma 7.(v), we know that g i (y) = g i (y) = t for every y ∈Γ i (t). Hence for t 1 = t 2 , it follows thatΓ i (t 1 ) ∩Γ i (t 2 ) = ∅. Therefore, there can only be countably many t ∈ R such thatΓ i (t) = ∅. For all other values, we see that Γ i (t) is a graph of a function over P . We denote this function by f i ( · , t).
We extend f i arbitrarily to the remaining values of t.
If t is such thatΓ i (t) = ∅, then Lemma 10 shows that for every y ∈ Γ i (t), the set Γ i (t) is between the cones y + (0, ∞) n and y − (0, ∞) n . It follows that f i ( · , t) is Lipschitz continuous.
Next we employ an approximation argument in conjunction with Lemma 11. Using a standard mollifier, we can find a sequence of smooth, uniformly bounded functions v k : R n → R such that v k → u locally uniformly as k → ∞ and [1, whenever Ω ⊆ R n is an open, bounded set with |D∇u|(∂Ω) = 0. It is then easy to modify v k such that in addition, it satisfies a j · ∇v k > −1 in R n for every j ∈ Z n+1 . Hence Lemma 11 applies to v k .
From the above convergence, it follows that for any sequence of points x k ∈ graph(v k ), if x k → x as k → ∞, then x ∈ graph(u). If we define φ k as in Lemma 11, then for any fixed t ∈ R, the functions φ k ( · , t) are uniformly bounded in C 0,1 (P ∩ B R (0)) for any R > 0. Hence there is a subsequence that converges locally uniformly. If t is such that Γ i (t) is the graph of f i ( · , t), then it is clear that the limit of any such subsequence must coincide with f i ( · , t).
Hence in this case, we have the locally uniform convergence φ k ( · , t) → f i ( · , t) as k → ∞. The second inequality in Lemma 11 implies that lim sup k→∞ˆR −RˆP ∩BR(0) Therefore, for almost every t ∈ (−R, R), there exists a subsequence (φ k ℓ ( · , t)) ℓ∈N converging to f i ( · , t) locally uniformly and such that lim sup ℓ→∞ˆP ∩BR(0) We conclude that f i ( · , t) ∈ BV 2 loc (P ) for almost all t ∈ R. We finally need to show that∇f i (p, t) ∈ {b 1 , . . . , b n } for almost every t ∈ R and H n−1 -almost every p ∈ P .
Consider the function w i : R n → R with Then for every t ∈ R, Note further that F i coincides up to an H n -null set with {x ∈ R n : ∇w i (x) = 0}. Let Z ⊂ R n denote the set of all points where u is not differentiable. By Rademacher's theorem, this is an H n -null set. Hence the coarea formula gives In particular, for almost all t ∈ R, is an H n−1 -null set, too. Therefore, for H n−1almost all y ∈ Γ i (t), the unique point x ∈ R with Φ i (U (x)) = y t belongs to R n \ Z and satisfies ∇u(x) ∈ A \ {a i }.
To put it differently, for almost every t ∈ R, the following holds true: for H n−1 -almost every p ∈ P the derivative of u exists at the point Θ(p, t) = tν i + n k=1 (p k + f i (p, t)σ k )ν i+k and belongs to A \ {a i }. Furthermore, we know that f i ( · , t) is differentiable at H n−1 -almost every p by Rademacher's theorem. At a point p ∈ P where both statements hold true, we can differentiate the equation (The right-hand side is the (n + 1)-st component of because p ∈ P and by the definition of σ.) For any ̟ ∈ P , we thus obtain If ∇u(Θ(p, t)) = a j for some j = i, then this simplifies to Hence We therefore conclude that∇f i (p, t) = b j−i at such a point.

Proof of Lemma 11
In this section we give the postponed proof of Lemma 11. To this end, we first need another lemma.
Proof of Lemma 11. First we note that by the assumptions on v, the intersection of graph(v) with the hyperplane x ∈ R n+1 : x · ν i = t is a smooth (n − 1)-dimensional manifold for every t ∈ R. Furthermore, the function φ is smooth. If we define Ξ : for p ∈ P and s, t ∈ R, then φ is characterised by the condition that for all t ∈ R and p ∈ P . Hence v Ξ(p, φ(p, t), t) = Ξ n+1 (p, φ(p, t), t).
It follows that |̟ ·∇φ(p, t)| ≤ √ n|̟|, and inequality (5) is proved. In order to prove the second statement of Lemma 11, we need to differentiate (7) again with respect to p. We write Λ : M for the Frobenius inner product between two matrices Λ and M . We also drop the arguments (p, t) in the derivatives of φ and in Θ. Then for all ̟, ξ ∈ P , : ∇ 2 v(Θ).
As we have already seen that |∇φ| ≤ √ n, it follows that there is a constant C 1 = C 1 (n) such that .
We also note that the map Θ is injective. Given R > 0, we therefore computê It remains to examine the set Θ((P ∩ B R (0)) × (−R, R)). Recall that we have the assumption sup R n |v| ≤ M in Lemma 11. Thus (6) implies that for all p ∈ P ∩B R (0) and all t ∈ (−R, R). Thus (9) implies the second inequality of Lemma 11.

Proof of Theorem 4
In this section we combine the previous results to prove the second main theorem. We first consider a function u ∈ BV 2 loc (R n ) ∩ L ∞ (R n ) such that graph(u) is close to the graph of λ i ∧ λ j or λ i ∨ λ j in a cube in R n+1 with edges parallel to ν 1 , . . . , ν n+1 . We will give a condition which implies that such a function actually coincides with λ i ∧ λ j or λ i ∨ λ j up to a constant in part of the domain.
For i, j ∈ Z n+1 with i = j and for r, R > 0, we define Again we consider the map U : R n → R n+1 with U (x) = ( x u(x) ) for x ∈ R n . The following is the key statement for the proof of Theorem 4.
Proposition 13. Let n ∈ N. For any δ > 0 there exist ǫ > 0 with the following properties. Let i, j ∈ Z n+1 with i = j. Suppose that |u(0)| ≤ ǫ and either Then If, in addition, then there exist α, β ∈ R such that or Before we can prove Proposition 13, we need a few more lemmas. First we need some more information on the functions f i from Proposition 8. Recall that f i ( · , t) ∈ BV 2 loc (P ) for almost all t ∈ R. Given i ∈ Z n+1 and given t ∈ R such that f i ( · , t) ∈ BV 2 loc (P ), let D ′ i (t) ⊆ P denote the approximate jump set of∇f i ( · , t). Thus this set is defined analogously to E ′ , but for the function∇f i ( · , t) instead of u. Furthermore, we set . Hence for any t 1 , t 2 ∈ R and any Borel set Ω ⊆ R n , t 2 )) .
Proof. Let p ∈ P and t ∈ R. Set If x ∈ F * , then Proposition 5 implies that graph(u) coincides with a hyperplane in a neighbourhood of x. If that hyperplane is perpendicular to ν i , then p + f i (p, t)σ ∈Γ i (t) and t belongs to the null set identified in Proposition 8.
Otherwise, the function f i ( · , t) is affine near p, and hence Φ i (x) cannot belong to D † i (t) × {t}. This implies the first claim. The second claim is now a consequence of the coarea formula [1, Theorem 2.93].
Proof. Consider the projection Π : R n+1 → R n given by Π(y) = y for y ∈ R n . Set Ψ i = Π • Φ i . Then for j ∈ Z n+1 with j = i and for x ∈ F * j , it is clear that J F * Ψ i (x) = 0. Hence the area formula gives H n (Ψ i (F * j )) = 0. This means that for H n−1 -almost every z ∈ R n−1 , for all j = i. Furthermore, since E * is an H n−1 -rectifiable set and H n−1 (N * ) = 0, we also know that for H n−1 -almost every z ∈ R n−1 , and Consider a point z ∈ R n−1 such that (16), (17), and (18) hold true. Recall that by Lemma 7, a point y ∈ R n+1 belongs to Φ i (graph(u)) if, and only if, g i (y) ≤ y n+1 ≤ g i (y). Also recall that From (16)-(18) we therefore infer that for H 1 -almost all y ∈ L z , Consider y ∈ Φ i (F * i ) with y ∈ L z . Then, setting x = Φ −1 i (y), we have the locally uniform convergence u x,ρ → λ i as ρ ց 0. Hence for any compact set K ⊆ R n+1 and any ǫ > 0 there exists ρ 0 > 0 such that for all ρ ∈ (0, ρ 0 ]. Recall that e 1 , . . . , e n are the standard basis vectors in R n . It follows that there exists r 0 > 0 such that for all r ∈ (0, r 0 ], g i (y ± re k ) − g i (y) ≤ rǫ and g i (y ± re k ) − g i (y) ≤ rǫ and |g i (y) − g i (y)| ≤ rǫ. Thus ∂ ∂y k g i (y) = 0 and ∂ ∂y k g i (y) = 0 and g i (y) = g i (y). Since this is true for H 1 -almost all y ∈ L z , Lemma 7.(vi) implies that for all y ∈ L z . If (19) holds for all y ∈ L z , then we immediately conclude that g i and g i are constant and coincide on L z , i.e., we have the first alternative from the statement of the lemma. If there exists y ∈ L z such that (19) does not hold true, then by the above observations, we know that holds in fact for all t ∈ [g i (y), g i (y)]. Moreover, because (19) still holds true almost everywhere on L z , there exists a sequence (ỹ m ) m∈N in L z such that y = lim m→∞ỹm and such that (19) holds for everyỹ m . We may then choosẽ . Extracting a subsequence if necessary, we may assume that y n+1 = lim m→∞tm exists. Set y = ( y yn+1 ). Then Φ −1 i (y) belongs to the boundary of F * i relative to graph(u). Proposition 5 implies that F * i is an open set relative to graph(u), and its relative boundary is contained in E * ∪ N * . Because of (18), it follows that Φ −1 i (y) ∈ E * . Moreover, (20) implies that Thus y has the properties from the second alternative in the statement.
Lemma 16. Let i ∈ Z n+1 . Suppose that G ⊆ R n is a connected set such that G ∩ Γ i (t) = ∅ for all t ∈ (−1, 1). Then either g i (y) ≥ 1 for all y ∈ G or g i (y) ≤ −1 for all y ∈ G.
Because g i is upper semicontinuous by Lemma 7, this is a closed set relative to G. Moreover, if y ∈ H t , it follows that g i (y) ≥ g i (y) ≥ 1, because G ∩ Γ i (t ′ ) = ∅ for all t ′ ∈ (−1, 1). By the lower semicontinuity of g i , this means that there exists ρ > 0 such that g i ≥ g i ≥ t in B ρ (y). Hence H t is also open relative to G. Since G is connected and y 0 ∈ H t , it follows that H t = ∅. This is true for all t ∈ (−1, 1), so g i (y) ≤ −1 for all y ∈ G.
We now have everything in place for the proof of Proposition 13.
Proof of Proposition 13. We use induction over n. The statement is clear for n = 1. We now assume that n ≥ 2 and the statement holds true for n − 1. For simplicity, we assume that i = 1 and j = 2. We also assume that (10) holds true; the proof is similar under the assumption (11).