Stable Norm in Variational Models

• Stable norm is defined as the limit of normalized energies of plane-like minimizers, capturing the effective anisotropy from microscopic periodic structures.
• It is characterized via a cell formula and Γ-convergence, linking nonlocal energies to local anisotropic perimeter functionals in variational problems.
• The concept connects discrete and continuum models, providing practical insights for homogenization theory and the study of minimal surfaces in heterogeneous media.

A stable norm is a homogenized surface tension functional that arises in the study of variational problems with periodic media, particularly in the context of anisotropic interfacial energies, nonlocal perimeters, and homogenization theory. In periodic perimeter minimization problems, the stable norm encodes the macroscopic, or effective, anisotropy resulting from the underlying microscopic periodic structure. It is defined as the limit of appropriately normalized energies of planelike minimizers in increasingly large domains, yielding a norm that governs the large-scale behavior of minimal surfaces subjected to periodic heterogeneity. The stable norm also appears as the limiting interfacial energy in nonlocal variational models, particularly under Γ-convergence to local anisotropic perimeter functionals.

1. Definition and Variational Framework

Consider a continuous periodic anisotropy $F(x,\nu)$ on the flat torus $$T = \mathbb{R}^n/\mathbb{Z}^n \times (\mathbb{R}^n\setminus\{0\})$$, which is one-homogeneous, convex in $\nu$, and uniformly elliptic. For a set of finite perimeter $E \subset \mathbb{R}^n$, and an open $A$, the anisotropic perimeter is defined by

$$E(E,A) = \int_{\partial^* E \cap A} F(x, \nu_E) \, d\mathcal{H}^{n-1}(x).$$

Under this setting, the stable norm $\phi:\mathbb{R}^n \rightarrow [0,\infty)$ is associated with the energy density per unit area in direction $p$ for "plane-like minimizers." The cell formula characterizes the stable norm variationally as

$\phi(p) = \min_{v \in BV(T)} \int_T F(x, p + Dv),$

defining a convex, one-homogeneous function. Equivalently, the stable norm can be expressed as the limit of rescaled energies of minimizers in cubes orthogonal to $p$,

$$\phi(p) = \lim_{R \to \infty} (\omega_{n-1}R^{n-1})^{-1} E(E, B_R),$$

where $E$ is a plane-like minimizer in direction $p$ (Chambolle et al., 2012).

In the nonlocal regime, the stable norm is constructed by taking the limit of normalized nonlocal energies of planelike minimizers in periodic media. For a kernel $K$ and periodic forcing $g$, and a planelike minimizer $E_p$, the stable norm is defined as

$$\phi(p) = \lim_{R \to \infty} R^{1-n} F(E_p, Q^p_R),$$

with $Q^p_R$ a cube orthogonal to $p$ (Dipierro et al., 13 Jan 2026).

2. Planelike Minimizers and Existence

A recurring concept in the construction of the stable norm is the existence of planelike minimizers. These minimizers are sets (or functions) whose interfaces or transition layers remain at a bounded distance from a reference hyperplane in a prescribed direction.

The existence theorem for planelike minimizers states that for every nonzero $p \in \mathbb{R}^n$, there exists a class-A minimizer $E$ such that

$$\{ x \cdot (p/|p|) > a+M \} \subset E \subset \{ x \cdot (p/|p|) > a - M \},$$

for some $M$ independent of $a$, and $\partial E$ is connected (Chambolle et al., 2012). In the nonlocal periodic setting, analogous results ensure that for each $p$ and for sufficiently small periodic forcing, there are minimizers whose boundary is constrained within a constant-width slab orthogonal to $p$ (Dipierro et al., 13 Jan 2026, Cesaroni et al., 2017, Cozzi et al., 2018, Cozzi et al., 2015, Cozzi et al., 2016).

Plane-like minimizers are constructed via constrained minimization in finite slabs, limiting procedures, and compactness, often coupled with Birkhoff ordering-type arguments to control oscillations and ensure existence for both rational and irrational directions.

3. Properties: Convexity, Differentiability, and Foliation

The stable norm exhibits strict convexity. Under the regularity and ellipticity assumptions on $F$, the map $p \mapsto \phi(p)^2$ is strictly convex (Chambolle et al., 2012).

Differentiability properties are governed by dynamical foliations of the torus (or the corresponding periodic cell). In totally irrational directions (i.e., $q \cdot p \neq 0$ for all $q \in \mathbb{Z}^n\setminus\{0\}$), the stable norm is differentiable at $p$: the subgradient $\partial \phi(p)$ is a singleton, and correctors (minimizers in the cell formula) are unique up to additive constants. For rational directions, the differentiability of $\phi$ at $p$ is equivalent to the property that the family of plane-like minimizers satisfying a strong Birkhoff property foliates the torus without gaps. If gaps occur (i.e., the corresponding lamination has nontrivial heteroclinic connections), then $\phi$ fails to be differentiable at $p$. This dichotomy is fundamental in homogenization theory and is established via calibration arguments and the analysis of cell problems (Chambolle et al., 2012).

4. Nonlocal Perimeters, Γ-convergence, and Effective Anisotropy

In nonlocal variational models, stable norms arise as the limit densities in homogenization processes. For functionals of the form

$$F_\epsilon(E, \Omega) = \iint_{\Omega_\sharp} \frac{1}{2} |\chi_E(x) - \chi_E(y)| K_\epsilon(x, y) dx dy + \int_{\mathcal Q(\Omega)_\epsilon \cap E} g_\epsilon(x) dx,$$

with suitably scaled kernel $K_\epsilon$ and external forcing $g_\epsilon$, one shows that as $\epsilon \to 0$, the functionals $\Gamma$-converge in the $L^1_{\operatorname{loc}}$ topology to the local anisotropic perimeter

$$F(E) = \int_{\partial^* E} \phi(\nu_E) d\mathcal{H}^{n-1}(x),$$

where $\phi$ is the stable norm derived from the nonlocal model (Dipierro et al., 13 Jan 2026). This result provides the rigorous identification of the effective surface tension in the homogenized (macroscopic) limit, with the anisotropy tensor determined by the underlying periodic medium or kernel.

5. Examples, Applications, and Explicit Computations

Applications and explicit computations include isotropic and anisotropic lattice examples, translation-invariant and non-translation-invariant cell problems, and G-closure phenomena for interfacial energies (Chambolle et al., 2012, Dipierro et al., 13 Jan 2026). In isotropic settings with $K(x,y) = |x-y|^{-n-2s}$ and $g \equiv 0$, the stable norm reduces to a constant, recovering the classical perimeter. In examples with nontrivial periodic forcing, the Wulff shape and curvature properties are determined by the interplay between the external field $g$ and the kernel $K$.

Concrete computation of the stable norm is often accomplished by solving a one-dimensional cell problem, minimizing $F(E_p, Q^p_R)$ for large $R$, or via direct numerical approximation.

6. Regularity, Uniqueness, and Links to Classical Theory

Regularity and uniqueness of minimizers in the stable norm context remain subtle. Uniqueness of planelike minimizers is generally not guaranteed; minimal or maximal envelopes among global minimizers may be identified, but full uniqueness is exceptional (Cesaroni et al., 2017, Cozzi et al., 2015). Regularity of the interface is limited: persistence of nonlocal oscillations beneath a characteristic scale may preclude $C^{1,\alpha}$ regularity theories applicable to classical minimal surfaces.

A fundamental asymptotic passage is the convergence, as the nonlocality parameter vanishes, of the stable norm and associated minimizers to those of the classical perimeter. Thus, the stable norm may be seen as interpolating, in various regimes, between local (classical) and nonlocal (fractional) geometric variational problems (Cesaroni et al., 2017, Cozzi et al., 2018).

7. Connections to Discrete Models and Future Directions

There exists a direct bridge between discrete long-range Ising models and stable norms via the continuum limit and $\Gamma$-convergence. Minimizers of Ising Hamiltonians with periodic or random coefficients yield, in the scaling limit, minimal surfaces for a nonlocal perimeter functional, and the corresponding stable norm. Extensions to random or almost-periodic environments, study of antiferromagnetic perturbations, and quantitative homogenization for nonlocal geometries remain open research directions (Cozzi et al., 2016).

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References

• "Plane-like minimizers and differentiability of the stable norm" (Chambolle et al., 2012)
• "Non-local planelike minimizers and $Γ$-convergence of periodic energies to a local anisotropic perimeter" (Dipierro et al., 13 Jan 2026)
• "Minimizers for nonlocal perimeters of Minkowski type" (Cesaroni et al., 2017)
• "Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium" (Cozzi et al., 2015)
• "Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media" (Cozzi et al., 2018)
• "Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces" (Cozzi et al., 2016)