Minimizers of anisotropic perimeters with cylindrical norms

We study various regularity properties of minimizers of the $\Phi$--perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

In this paper we are interested in regularity properties of minimizers of the anisotropic perimeter of E in Ω, and of the related area-type functional Here Ω ⊆ R n+1 is an open set, Φ : R n+1 → [0, +∞) is a norm (called anisotropy), Φ o is its dual, E ⊂ R n+1 is a set of locally finite perimeter, ∂ * E is its reduced boundary, ν E is the outward (generalized) unit normal to ∂ * E, and H n is the n -dimensional Hausdorff measure in R n+1 . On the other hand, Ω ⊆ R n , v belongs to the space BV loc ( Ω) of functions with locally bounded total variation in Ω, and Dv is the distributional derivative of v. When Ω = Ω × R the two functionals coincide provided E is cartesian, i.e. E is the subgraph sg(v) ⊂ Ω × R of the function v ∈ BV loc ( Ω) (see (4.1)). Anisotropic perimeters appear in many models in material science and phase transitions [21,37], in crystal growth [7,8,12,13,39,3], and in boundary detection and tracking [15]. Functionals like G Φ o , having linear growth in the gradient, appear quite frequently in calculus of variations [20,9,6].
The one-homogeneous case is particularly relevant, since it is related to the anisotropic total variation functional (1.1) a useful functional appearing, for example, in image reconstruction and denoising [34,16,17,5,28].
Here ϕ : R n → [0, +∞) is a norm, and its dual ϕ o is typically the restriction of Φ o on the "horizontal" R n . Minimizers of P Φ have been widely studied [37,4]; in particular, it is known [10,2] that if Φ 2 is smooth and uniformly convex, (boundaries of) minimizers are smooth out of a "small" closed singular set. In contrast to the classical case, where perimeter minimizers are smooth out of a closed set of Hausdorff dimension at most n − 7, the behaviour of minimizers of anisotropic perimeters is more irregular: for instance, there exist singular minimizing cones even for smooth and uniformly convex anisotropies in R 4 [29]. Referring to functionals of the form (1.1), we recall that, if n ≤ 7, Hölder continuity of minimizers for the image denoising functional [34], consisting of the Euclidean total variation T V plus the usual quadratic fidelity term, has been studied in [14]. In [26] such result is extended to the anisotropic total variation T V ϕ .
One of the remarkable results in the classical theory of minimal surfaces is the classification of entire minimizers of the Euclidean perimeter P : if n ≤ 6 the only entire minimizers are hyperplanes, while for n = 7 there are nonlinear entire minimizers (see for instance [20,Chapter 17] and references therein); in the cartesian case (sometimes called the non parametric case), this is the well-known Bernstein problem. In the anisotropic setting, to our best knowledge, only a few results are available: entire minimizers in R 2 are classified in [32], and minimizing cones in R 3 for crystalline anisotropies are classified in [38]. In [23,35] the authors show that if n ≤ 2 and Φ 2 is smooth, the only entire cartesian minimizers are the subgraphs of linear functions (anisotropic Bernstein problem), and the same result holds up to dimension n ≤ 6 if Φ is close enough to the Euclidean norm [35]. However, the anisotropic Bernstein problem seems to be still open in dimensions 4 ≤ n ≤ 6, even for smooth and uniformly convex norms (see [33] for recent results in this direction).
The above discussion shows the difficulty of describing perimeter minimizers in the presence of an anisotropy; it seems therefore rather natural to look for reasonable assumptions on Φ that allow to simplify the classification problem. A possible requirement, which will be often (but not always) assumed in the sequel of the paper, is that Φ is cylindrical over ϕ, i.e. Φ( ξ, ξ n+1 ) = max{ϕ( ξ), |ξ n+1 |}, ( ξ, ξ n+1 ) ∈ R n+1 .
(1.2) Despite its splitted expression, a cylindrical anisotropy is neither smooth nor strictly convex, and this still makes the above mentioned classification rather complicated. For instance, in Examples 2.7 and 2.9 we show that there exist singular cones minimizing P Φ in any dimension n ≥ 1. Moreover, while it can be proved that if horizontal and vertical sections of E are minimizers of P ϕ and P respectively then E is a minimizer of P Φ (Remark 2.6), in general sections of a minimizer of P Φ need not satisfy this minimality property (Examples 2.8 and 2.9). These phenomena lead us to investigate the classification problem under some simplifying assumptions on the structure of minimizers. We shall consider two cases: cylindrical minimizers (Definition 3.1), and cartesian minimizers (Definition 4.1), the latter being our main interest. Cylindrical minimizers of P Φ are studied in Section 3: in particular, in Example 3.6 we classify all cylindrical minimizers of P Φ when n = 2 and the unit ball B Φ of Φ (sometimes called Wulff shape) is a cube. Cartesian minimizers are studied in Sections 4, 5 and 6. In Section 4 we investigate the relationships between cartesian minimizers of P Φ and minimizers of G Φ o , provided Φ is partially monotone (Definition 4.4). In Theorem 4.6 we show that the subgraph of a minimizer of G Φ o is also a minimizer of P Φ among all perturbations not preserving the cartesian structure. In particular, for Φ satisfying (1.2) the subgraph E of some function u : Ω → R is a cartesian minimizer of P Φ in Ω × R if and only if u is a minimizer of T V ϕ .
Sections 5 and 6 contain our main results, valid under the assumptions that Φ is cylindrical over ϕ and E is cartesian.
In Theorem 5.8 (see also Corollary 5.12) we prove the following Bernstein-type classification result: if eiter n ≤ 7 and ϕ is Euclidean, or if n = 2 and ϕ o is strictly convex, then any entire cartesian minimizer of P Φ in R n+1 (i.e. the subgraph of a minimizer of T V ϕ ) is the subgraph of the composition of a monotone function on R with a linear function on R n . We notice that this result is sharp: if n = 8, there are entire cartesian minimizers of P in R 9 which cannot be represented as the subgraph of the composition of a monotone and a linear function (see Remark 5.11).
In view of our assumptions, also the regularity results of Section 6 are concerned with the anisotropic total variation functional. For our purposes, it is useful to remark that, even if the anisotropy ϕ is smooth and uniformly convex, in general minimizers of T V ϕ are not necessarily continuous. In contrast, we remark that minimizers of T V ϕ with continuous boundary data on bounded domains are continuous, see [22,26,25]. Nevertheless, in Theorems 6.2 and 6.4 we show that, if ϕ 2 ∈ C 3 is uniformly convex, then the boundary of the subgraph of a minimizer of T V ϕ is locally Lipschitz (that is, locally a Lipschitz graph) out of a closed singular set with a suitable Hausdorff dimension depending on ϕ . As observed in Remark 6.3, for ϕ Euclidean these statements are optimal, while the statement is false already in dimension n = 2 for ϕ the square norm.

NOTATION AND PRELIMINARIES
In what follows n ≥ 1 , Ω ⊆ R n+1 and Ω ⊆ R n are open sets. BV (Ω) (resp. BV loc (Ω) ) stands for the space of functions with bounded (resp. locally bounded) variation in Ω [6]. The characteristic function of a (measurable) set E ⊂ Ω is denoted by χ E ; we write E ∈ BV (Ω) (resp. E ∈ BV loc (Ω) ) when χ E ∈ BV (Ω) (resp. χ E ∈ BV loc (Ω) ). Similar notation holds in Ω. P (E, A) denotes the Euclidean perimeter of the set E in the open set A. Recall that the perimeter of E ∈ BV loc (Ω) does not change if we change E into another set in the same Lebesgue equivalence class; henceforth we shall always assume that any set E coincides with its points of density one [6,19]. The outward generalized unit normal to the reduced boundary ∂ * E of E ∈ BV loc (Ω) is denoted by ν E . We often use the (2.1) Unless otherwise specified, in the sequel we take m ∈ {n, n + 1} .
Definition 2.2 (Cylindrical and conical norms). We say that the norm Φ : where ϕ : R n → [0, +∞) is a norm. We say that Φ : Notice that if Φ is cylindrical over ϕ then Φ o is conical over ϕ o , and vice-versa. For any E ∈ BV loc (O) and for any A ∈ A c (O) we define [5] the Ψ -perimeter of E in A as It is known [5] that The following example is based on a standard calibration argument 1 .
1 See for instance [1] for some definitions, results and references concerning calibrations.
Example 2.5 (Parallel planes). Let n = 2, and let Φ : R 3 → [0, +∞) be cylindrical over the Euclidean norm. Given a < b consider E = {(x, t) ∈ R 3 : a < t < b}. Then E is not a minimizer of P Φ in R 3 . Indeed, it is sufficient to compare E with the set E \ C, obtained from E by removing a sufficiently large cylinder Then P Φ (E) is reduced by 2πR 2 (the sum of the areas of the top and bottom facets of C ), while it is increased by the lateral area 2π(b−a)R of C. Hence, for R > 0 sufficiently large, (2.8) is not satisfied. Notice that the horizontal sections of E are either empty or a plane, which both are minimizers of the Euclidean perimeter in R 2 .
Remark 2.6. Suppose that Φ : R n+1 → [0, +∞) is cylindrical over ϕ. Assume that E ∈ BV loc (Ω) has the following property: for almost every t ∈ R the set E t (horizontal section) is a minimizer of P ϕ in Ω t and for almost every x ∈ R n the set E x (vertical section) is a minimizer of Euclidean perimeter in Ω x . Then by Remark A.2 we get that E is a minimizer of P Φ in Ω.
Example 2.7. For any l, γ ∈ R we define the cones 3 in R n+1 From Example 2.4 and Remark 2.6 it follows that the following sets are minimizers of P Φ in R n+1 provided that Φ satisfies (2.6): Figure 1). b) The union of C (n) 1 (l 1 , γ) and the rotation of C (n) 2 (l 2 , γ) around the vertical axis x n+1 of α radiants (see Figure 2). 2 Up to sets of zero H m−1 -measure. 3 A set E ⊆ R m is a cone if there exists x0 ∈ ∂E such that for any x ∈ E and λ > 0 it holds x0 + λ(x − x0) ∈ E.
2 (0, 0) in Example 2.7(b). Notice that C t for t = 0 is not a minimizer of the Euclidean perimeter in R 2 ; however, this does not affect the minimality of C.
In general, a minimizer of P Φ in Ω for a cylindrical Φ, need not satisfy the minimality property of horizontal sections in Remark 2.6.
For a similar reason, taking into account the term A |D x 1 χ F | we may assume that E ∩ ∂ * F lies on two horizontal parallel lines at distance ε ∈ (0, γ). Then by definition of ϕ -perimeter This implies that E is a minimizer of P ϕ in Ω. Notice that every horizontal section of E is (0, l), which is not a minimizer of the perimeter in R.
Notice that every horizontal section of E is a translation of the strip (0, l) × R, which is not a minimizer of P ϕ in R 2 according to Example 2.5.
is a minimizer of P Φ in R 2 even though in case (c) for any t ∈ (0, γ), the horizontal section E t is not a minimizer of the perimeter in R (see (2.9)).
Indeed, if l ≥ 0 ≥ γ then E satisfies the property in Remark 2.6. If l = 0 and γ > 0, then R 2 \ E is union of two disjoint cones satisfying property stated in Remark 2.6. Thus, in both cases E is a minimizer of P Φ in R 2 .
On the other hand, it is not difficult to see that among all perturbations of E involving both componenents, the best one is obtained by inserting an horizontal strip as in Figure 3. However, because of the assumption 0 < γ ≤ l, this perturbation has larger Φ -perimeter than E. Consequently, E is a minimizer of P Φ in R 2 .

CYLINDRICAL MINIMIZERS
Let Φ be a norm on R n+1 and Ω = Ω × R.
The aim of this section is to characterize cylindrical minimizers of P Φ . The idea here is that the (Euclidean) normal to the boundary of a cylindrical minimizer is horizontal, and therefore what matters, in the computation of the anisotropic perimeter, is only the horizontal section of the anisotropy. For this reason it is natural to introduce the following property, which informally requires the upper (and the lower) part of the boundary of the Wulff shape to be a generalized graph (hence possibly with vertical parts) over its projection on the horizontal hyperplane R n × {0}.
Definition 3.2 (Unit ball as a generalized graph in the vertical direction). We say that the boundary of the unit ball B Φ of the norm Φ : But ∂B Φε is a generalized graph in the vertical direction. Indeed, consider the straight line passing through (1, 0) and parallel to ξ 2 -axis. This line does not has a unit ball the boundary of which is not a generalized graph in the vertical direction, since Φ(2, 0) = 2 > √ 3 = Φ(2, −1).
The following assertions hold: Then E∆F m ⊂⊂ A × I m+1 ⊂⊂ Ω × R and, by minimality, . As a consequence, , and assertion (a) follows.
and since E is a minimizer of P ϕ in Ω, using (3.1) and (A.1) we get and assertion (b) follows.
Example 3.6 (Characterization of cylindrical minimizers for a cubic anisotropy). Proposition 3.4 allows us to classify the cylindrical minimizers of P Φ for suitable choices of the dimension and of the anisotropy. Take n = 2, Ω = R 2 , and let in particular, ∂B Φ is a generalized graph in the vertical direction and B ϕ is the square [−1, 1] 2 in the (horizontal) plane. The minimizers of P ϕ are classified as follows [32, Theorems 3.8 (ii) and 3.11 (2)]: the infinite cross C = {|x 1 | > |x 2 |} and its complement, the subgraphs and epigraphs S of monotone functions of one variable, and suitable unions U of two connected components, each of which is the subgraph of a monotone function of one variable. Then Proposition 3.4 (b) implies that are the only cylindrical minimizers of P Φ in R 3 . The same result holds if B ϕ is a parallelogram centered at the origin, and ∂B Φ is any generalized graph in the vertical direction such that B ϕ = B Φ ∩ {ξ 3 = 0}.

(4.4)
Supposing that the claim is true, from (4.4) and from Lemma A.7 we deduce Then by the minimality of u and (4.1) we get Let us prove our claim. Since sg(u)∆F ⊂⊂ A, we have belongs to L 1 loc ( Ω) and the sequence {v h } converges pointwise to v ∈ BV loc ( Ω) as h → +∞. To show that u − v is compactly supported in A it is enough to take A ∈ A c ( Ω) such that A ⊂⊂ A and sg(u)∆F ⊂⊂ A × (−M, M ), and to observe that since sg(u) Hence, from (4.5), for almost every x ∈ Ω we get Let us fix ψ ∈ C 1 c ( Ω) and 1 ≤ j ≤ n. Then, using Ω D x j ψ(x)dx = 0, the dominated convergence theorem (see [20,27] for more details) and (4.6) we find Hence for any A ∈ A c ( Ω) and η ∈ C 1 c ( A; B ϕ 1 ) one has

Since η is arbitrary, the definition of
Being |D t χ F | a counting measure, we have [27] Moreover, one checks that

CLASSIFICATION OF CARTESIAN MINIMIZERS FOR CYLINDRICAL NORMS
The aim of this section is to give a rather complete classification of entire cartesian minimizers, supposing the norm Φ cylindrical. As explained in the introduction, this case covers, in particular, the study of minimizers of the total variation functional. We start with a couple of observations.
On the other hand, (5.1) is expected to hold not for all non cylindrical norms Φ. For example, let Φ be Euclidean, n ≥ 8 and u : R n → R be a smooth nonlinear solution [11] of the minimal surface equation div Indeed, otherwise λu solves the minimal surface equation, hence 0 =λdiv ∇u If ∆u|∇u| 2 is not identically zero, we get λ(1 − λ 2 ) = 0, a contradiction. Further properties of cartesian minimizers are listed in the following proposition, which in particular (when ϕ is Euclidean) asserts some properties of minimizers of the total variation functional [17].

Proposition 5.3 (Cartesian minimizers for cylindrical norms). Suppose that
is cylindrical over ϕ. The following assertions hold:  Clearly, assertion (e) generalizes (5.1) and (5.2). We also anticipate here that the converse of statement (f) is considered in Theorem 5.8 below.
Proof. The proof of (a) is the same as in [11,Theorem 1]

and (b) is immediate. (c) follows from (a) and (b), while (d) follows from the coarea formula
Let us prove (e). Without loss of generality assume that f is nondecreasing. Suppose first that f is Lipschitz and strictly increasing.
In the general case, it is sufficient to approximate f with a sequence of strictly increasing Lipschitz functions, and use Theorem 4.3. (f) follows from (e), since the linear function u 0 (x) = x · ζ, x ∈ Ω, is a minimizer of G Φ o in Ω.
Before proving the proposition, some comments are in order. Our assumptions on H 1 and H 2 exclude, in particular, that E is a "roof-like" cone (as the one depicted in Figure 4). More specifically, in case ν 1 = ν 2 , the inclusion ∂H 1 ∩ ∂H 2 ⊂ {t = 0} in (5.3) implies that the orthogonal complement to {t = 0} is contained in the span of the orthogonal complements of ∂H i , i.e. e n+1 ∈ span(ν 1 , ν 2 ).
Next, assumption (a) implies that ν 1 and ν 2 lie "on the same side" with respect to e n+1 , while assumption (b) implies that ν 2 lies between ν 1 and e n+1 (a condition not satisfied in Figure 4, and satisfied in Figure 5). We shall see in Example 5.6 that, if condition (b) is not satisfied, then E and F need not be minimizers.
If λ 2 = +∞, then by conditions (a) and (b) we have H 1 = H 2 = H, where H is the half-space whose outer unit normal is −( ν, 0). By Example 2.4 it follows that H = E = F is a minimizer of P Φ .
Then the cones E := H 1 ∩ H 2 and F := H 1 ∪ H 2 are not minimizers of P Φ . Let us prove the assertion for E, the statement for F being similar. The lines ∂H 1 , ∂H 2 and {t = −1} compose a nondegenerate triangle T ⊂ E with sides a 1 , a 2 , b > 0, b the horizontal side. For any A ∈ A c (R 2 ) with T ⊂⊂ A we have We shall need the following relevant result (see for instance [20,Theorem 17.3] and references therein).
Theorem 5.7. Let E be a minimizer of the Euclidean perimeter in R n . Then either n ≥ 8 or ∂ E is a hyperplane.
Set v(x) := f (x · ζ). By construction, we have {v > λ} = {u > λ} for a.e. λ ∈ R. It is easy to check that if w ∈ L 1 loc (R n ) then for a.e. x ∈ R n one has hence u = v almost everywhere on R n .
Remark 5.11. Assumption (a) of Theorem 5.8 is optimal in the sense that if n ≥ 8 there exist minimizers of G Φ o on R n which cannot be written as in (5.4 for any x ∈ R n .

LIPSCHITZ REGULARITY OF CARTESIAN MINIMIZERS FOR CYLINDRICAL NORMS
We recall from [31,Theorem 3.12] that if n = 2 and if ∂B ϕ either does not contain segments, or it is locally a graph in a neighborhood of its segments, then the graph of a minimizer of G Φ o in R 2 is locally Lipschitz. On the other hand, an example in [31,Sect. 4] shows that such a regularity result cannot be expected for a general anisotropy. More precisely, for Φ o cylindrical as in (2.6) with ϕ o ( ξ * ) = |ξ * 1 | + |ξ * 2 |, that example exhibits a function u ∈ M Φ o (R 2 ) such that the set of points where the boundary of sg(u) is not locally the graph of a Lipschitz function has positive H 2 -measure. We look for sufficient conditions on ϕ which exclude such pathological example.
Let us start with a regularity property of cartesian minimizers of G Φ o for cylindrical norms over the Euclidean norm, namely for Φ( ξ, ξ n+1 ) = max(| ξ|, |ξ n+1 |), which is exactly the case of the total variation functional.
Remark 6.5. In [31] it is proven that if n = 2, B ϕ is not a quadrilateral, and u is a minimizer of G Φ o in Ω, then the graph of u is locally Lipschitz around any point of ∂sg(u). Remark 6.6. Using the regularity result in [2,Theorem II.8], under the assumption that ϕ is uniformly convex, smooth and sufficiently close to the Euclidean norm, one can improve Theorem 6.4 by showing that Σ(u) has Hausdorff dimension at most n − 5.
Proposition A.1. Let E ∈ BV loc (Ω). Then for any A ∈ A c (Ω) where E t and E x are defined as (2.1), ν Et is a outer unit normal to ∂ * E t and ν Ex is a outer unit normal to ∂ * E x .