Plane-Like Minimizers and Differentiability of the Stable Norm

Abstract

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.

Appendices

Appendix A: A Discrete “Separation” Result

We show here the following result (see [12] for a slightly more complex proof):

Lemma A.1

Assume E is a set which satisfies the Birkhoff property, that is, for any q∈ℤ^d, either q+E⊆E or q+E⊇E. Then there exists p∈ℝ^d, |p|=1, such that for any q∈ℤ^d, q⋅p>0⇒q+E⊆E and (obviously) q⋅p<0⇒q+E⊇E. Moreover, p is unique, unless E+q=E for all q∈ℤ^d.

Proof

Let Z={q∈ℤ^d : q+E⊂E}, clearly, 0∈Z, Z+Z=Z (so that nZ⊂Z for any n≥0), and the Birkhoff property states that for any z∈ℤ^d, either z∈Z or −z∈Z. Without loss of generality we may therefore assume that e _i ∈Z for i=1,…,d, where (e _i ) is the canonical basis of ℝ^d.

We claim that either Z=ℤ^d, or the closed convex envelope of Z is not ℝ^d. In the latter case, since this envelope is a convex cone, it must be contained in a semispace, hence the result.

Assume, thus, that Z≠ℤ^d but any point in ℤ^d is in the convexification of Z. In particular, it must be that $-\sum_{i} e_{i}\not\in Z$, otherwise, for any p=(p ₁,…,p _d )∈ℤ^d we would obtain that given a nonnegative integer n≥−min _i p _i , p=−n∑ _i e _i +∑ _i (p _i −n)e _i is also in Z.

A consequence is that as soon as p _i <0 for all i, then $p\not\in Z$, otherwise p+∑ _i (−p _i −1)e _i =−∑ _i e _i ∈Z, which gives another contradiction.

Now, by assumption, for any ε>0 there exist $(p^{k})_{k=1}^{K}$, $(\theta^{k})_{k=1}^{K}$ with p ^k∈Z, θ ^k≥0, ∑ _k θ ^k=1 such that

$$\left| \sum_{k=1}^K \theta^k p^k + \sum_{i=1}^d e_i \right| < \varepsilon . $$

Possibly changing infinitesimally the θ ^k’s we can assume that they are rational, hence, θ ^k=n ^k/m for some integers n ^k≥0, m>0, ∑ _k n ^k=m.

Letting p=∑ _k n ^k p ^k∈Z, it follows that

$$\left| p + m\sum_{i=1}^d e_i \right| < m\varepsilon . $$

As a consequence, for each i, p _i <−m(1−ε)≤0 as soon as ε<1, which implies that $p\not\in Z$, a contradiction. Hence, the closed convex envelope of Z is strictly contained in ℝ^d. □

Appendix B: A Generic Uniqueness Result

In Theorem 4.23 we have shown that the minimizer of (6) is unique up to an additive constant, if the direction p is irrational. In addition, we show here that it is generically unique when p is rational, that is, we prove a geometric counterpart of Mañé’s conjecture [29].

For this we follow and adapt the proof of [14]. Contrary to the non-parametric case, it is no longer true that, if F is an admissible anisotropy and $f\in C^{\infty}(\mathbb{T})$, then F(x,p)+f(x)|p| is also an admissible anisotropy. Indeed, if inff≤−c ₀, the function F(x,p)+f(x)|p| is not coercive anymore. We will thus restrict ourselves to positive perturbations. For this reason we cannot directly use [14, Theorem 5] as in [16].

We will try to stay as close as possible to the notation of [14]. A set $\mathcal{O}$ is called a residual set if it is a countable intersection of dense open sets. In a complete metric space, by Baire’s theorem, this implies that $\mathcal{O}$ is itself dense.

Theorem B.1

For every rational vector p, there exists a residual set $\mathcal{O}(p)$ of $E:=C^{\infty}(\mathbb{T})\cap\{f\ge0\}$ such that for every $f\in \mathcal{O}(p)$, the minimizer of (6) for F+f(x)|p| is unique up to an additive constant.

Following an idea of Mather, we first rewrite (6) as a linear problem. Notice that, for $u\in \mathit{BV}(\mathbb{T})$, if we set $\mu_{u}:= |Du+p|\otimes\delta_{\frac{Du+p}{|Du+p|}}$ (which is a Radon measure on $\mathbb{T}\times \mathbb {S}^{d-1}$) then

$$\int_{\mathbb{T}} F(x,Du+p) = \int_{\mathbb{T}\times \mathbb {S}^{d-1}} F(x,\nu)\, d\mu_u $$

is linear in μ _u . Let $\widetilde{H}_{r}$ be the set of measures μ _u for $u\in \mathit{BV}(\mathbb{T})$ with total variation less than r. Let H _r be the weak-∗ closure of the convex hull of $\widetilde{H}_{r}$. By Banach–Alaoglu’s theorem, H _r is compact, convex, and metrizable. Let F be the space of Borel measures on $\mathbb{T}\times \mathbb {S}^{d-1}$, G be the space of Borel measures on $\mathbb{T}$, and K _r ⊂G be the metrizable compact convex set of Radon measures on $\mathbb{T}$ with total variation less than r. Then, if π:F→G is the projection on the first marginal, for every $\mu_{u}\in\widetilde{H}_{r}$, π(μ _u )=|Du+p|∈K _r and thus π(H _r )⊂K _r . Letting

$$\mathit{MA}(F,\mu) := \int_{\mathbb{T}\times \mathbb {S}^{d-1}} F(x,\nu)\, d\mu\quad\mu \in H_r , $$

we have that MA(L,⋅) is linear and continuous for the weak-∗ topology on H _r . If u is a minimizer of (6) for F+f with f∈E, we have

$$\left(\frac{1}{c_0}+|f|_\infty\right)|p| \ge \int_{\mathbb{T}} F(x,Du+p)+f(x)|Du+p| \ge c_0 |Du+p|(\mathbb{T}) $$

and thus for $r\ge(c_{0}^{-1}+|f|_{\infty})|p|/c_{0}$ the measure μ _u is the minimizer of MA(F+f,⋅) in $\widetilde{H}_{r}$. Since MA(F+f,⋅) is linear, the minimum over H _r and over $\widetilde{H}_{r}$ coincide. Hence, for every minimizing u, the measure μ _u is also a minimizer of MA(F+f,⋅) in H _r . We are thus going to prove that such minimizers in H _r are generically unique. Following [14], let $M_{H_{r}}(F):=\operatorname*{\arg\!\min}_{\mu\in H_{r}} \mathit{MA}(F,\mu)$ and $M_{K_{r}}(F):=\pi(M_{H_{r}}(F))$.

Proposition B.2

For every r>0 there exists a residual set $\mathcal{O}_{r}\subset E$ such that for every $f\in \mathcal{O}_{r}$, the set $M_{K_{r}}(F+f)$ is reduced to a single element.

Proof

Now let $\mathcal{O}(\varepsilon )$ be the set of points f∈E such that $M_{K_{r}}(F+f)$ is contained in a ball of radius ε in K _r . We will prove that the proposition holds for

$$\mathcal{O}_r:=\bigcap_{\varepsilon >0} \mathcal{O}(\varepsilon ). $$

Indeed, if $f\in \mathcal{O}_{r}$, and if $M_{K_{r}}(F+f)$ is not a singleton then it is a convex set of positive dimension which would not be included in a ball of radius ε for ε small enough, contradicting the hypothesis $f\in \mathcal{O}_{r}$. It is thus enough to prove that for every ε>0, the sets $\mathcal{O}(\varepsilon )$ are open and dense. □

The fact that $\mathcal{O}(\varepsilon )$ is open is a direct consequence of the continuity of the map

$$(f,\mu) \to \int_{\mathbb{T}\times \mathbb {S}^{d-1}} F(x,\nu)+f(x)|\nu|\, d\mu $$

which implies that for every open subset U of H _r , the set $\{f\in E \, :\, M_{H_{r}}(F+f)\subset U\}$ is an open set of E and similarly for $M_{K_{r}}(F+f)$. The density argument is more involved. Let w∈E; we want to prove that w is in the closure of $\mathcal{O}(\varepsilon )$. Repeating verbatim the proof of [14, Lemma 7] the following holds.

Lemma B.3

There exists an integer m and a continuous map T _m :K _r →ℝ^m

$$T_m(\eta) := \left( \int_{\mathbb{T}} w_1\, d\eta,\ldots, \int_{\mathbb{T}} w_m\, d\eta\right) $$

with w _i ∈E and such that

$$\forall\, x\in \mathbb{R}^m \quad\operatorname{diam} T_m^{-1}(x)<\varepsilon $$

where the diameter is taken for the distance on K _r .

Define the function Λ _m :ℝ^m→ℝ∪{+∞} as

$$\varLambda_m(x) := \min_{\stackrel{\mu\in H_r}{T_m\circ\pi(\mu )=x}} \mathit{MA}(F+w,\mu) $$

if x∈T _m (π(H _r )) and +∞ otherwise. For y=(y ₁,…,y _m )∈ℝ^m, let

$$M_m(y) := \operatorname*{\arg\!\min}_{x\in \mathbb{R}^m} \varLambda_m(x)+y\cdot x. $$

Then for $y\in \mathbb{R}^{m}_{+}$

$$M_{K_r}\Biggl(F+w+\sum_{i=1}^m y_i w_i\Biggr) \subset T_m^{-1}(M_m(y)). $$

Letting $\mathcal{O}_{m}:=\{y\in \mathbb{R}^{m} \, :\, M_{m}(y) \mbox{ is reduced to a point}\}$, we have from Lemma B.3

$$y\in \mathcal{O}_m \quad\mbox{ and }\quad y\in \mathbb{R}^m_+ \quad \Longrightarrow\quad w+\sum _{i=1}^m y_i w_i \in \mathcal{O}(\varepsilon ). $$

It is thus enough to prove that 0 can be approximated by positive vectors of $\mathcal{O}_{m}$.

For this, consider the convex conjugate of Λ _m ,

Since H _r is compact, it is a convex and finite-valued function which is then continuous on ℝ^m. Letting $\varSigma:= \{y\in \mathbb{R}^{m} \, : \, \operatorname{dim}\partial G_{m}(y)\ge1\}$ we have that $\operatorname{dim}\varSigma\le m-1$ (see [14, Appendix A] or [1]) and thus the complement of Σ is dense in ℝ^m. Since for every $y\in \mathbb{R}^{m}_{+}$ we have M _m (y)=∂G _m (−y), it follows that for every $y\in \mathbb{R}_{m}^{+}\cap\varSigma^{c}$ the set M _m (y) is reduced to a point, which proves the claim.

We can finally end the proof of Theorem B.1. Let $\mathcal{O}(p):=\bigcap_{r>0} \mathcal{O}_{r}$. Then by Baire’s theorem, $\mathcal{O}(p)$ is still a residual set of E. If $f\in \mathcal{O}(p)$ then for $r\ge (c_{0}^{-1}+|f|_{\infty})|p|/c_{0}$, the set $M_{K_{r}}(F+f)$ is reduced to a single element and if u and v are two different minimizers of (6) for F+f, it follows that |Du+p|=|Dv+p|. For s∈ℝ and E _s :={u+p⋅x>s}, we can construct as in Proposition 4.18 a minimizer $\widetilde{u}$ such that the level sets of $\widetilde{u} +p\cdot x$ correspond exactly to the projection Π(∂E _s ) of ∂E _s in the torus $\mathbb{T}$. Therefore, the measure |Du+p| is concentrated on Π(∂E _s ) and since on ∂E _s it holds that $\frac{Du+p}{|Du+p|}=\nu^{E_{s}}$, we find that $Du+p=D\widetilde{u} +p$ and hence Du is unique.  □

Remark B.4

From this uniqueness result, it can be easily proved that every plane-like minimizer with the Birkhoff property generically induces the same current. Since it is not the main focus of our work and it would consist in repeating the arguments in [16], we just sketch the proof. For $p\in \mathbb {S}^{d-1}$, u a minimizer of the cell problem, E a plane-like minimizer with the Birkhoff property, and $v\in C^{\infty }(\mathbb{T})$ a periodic vector field, define the currents T _u and T _E by

$$T_u(v):= \int_{\mathbb{T}} v\cdot(Du+p) \quad\mbox{and} \quad T_E(v):=\lim_{R\to+\infty} \frac{1}{\mathcal{H}^{d-1}(\partial B_R)} \int _{\partial^*E \cap B_R} v\cdot\nu^E \, d\mathcal{H}^{d-1}, $$

where the limit defining T _E exists arguing as in [15]. If Du is unique and E is recurrent then T _u =T _E . Now, for every plane-like minimizer E with the Birkhoff property and every q∈ℤ^d, we have T _E+k =T _E (see [16, Lemma 3.1]). Thus, letting $\widetilde{E}:= \bigcap_{q\cdot p>0} (E+q)$, it holds that $T_{\widetilde{E}}=T_{E}$ (see [16, Lemma 4.4]). Since $\widetilde{E}$ is recurrent, this implies that T _E =T _u and therefore every plane-like minimizer with the Birkhoff property induces the same current T _u .