Anisotropic Curvature Flow

• Anisotropic curvature flow is a geometric evolution process where interface velocity depends on local orientation and curvature, generalizing classical mean curvature flow.
• It employs PDE models and variational schemes to capture anisotropic effects, ensuring convexity preservation and establishing barrier methods for global bounds.
• This flow has significant applications in materials science, capillarity, and image processing, modeling phenomena such as crystalline behavior and droplet dynamics.

Anisotropic curvature flow describes a large and technically rich class of geometric evolution laws in which the velocity of hypersurfaces, curves, or interfaces is governed by a curvature functional that depends explicitly on the local orientation (“anisotropy”) of the interface normal. These flows generalize the classical (isotropic) mean curvature flow by incorporating orientation-dependent surface energies or kinetic effects, leading to PDEs and variational schemes governed by convex or crystalline norms. Within the anisotropic category, the flows exhibit profound distinctions in regularity, asymptotic behavior, singularity formation, and connection to both analytic and variational (minimizing movements) approaches.

1. Mathematical Formulation and Geometric Setting

The general evolution law for anisotropic curvature flow is

$V = A(\mathbf{n}) H + B(\mathbf{n}),$

where $V$ is the normal velocity, $H$ the scalar curvature, $\mathbf{n}$ the unit normal, and $A,B$ smooth, direction-dependent (“anisotropy”) functions on the sphere $S^{d-1}$ (Yuan et al., 2020). In the classical context, $A,B$ are constant and $V=H$ reduces to mean curvature flow (MCF). More commonly, $A(\mathbf{n})$ encodes the density for a surface energy, and $B(\mathbf{n})$ represents kinetic or drift effects.

The evolution equation for a graph $y = u(x,t)$ under this law, in a planar band domain, takes the explicit form

$$u_t = a(u_x)\frac{u_{xx}}{1 + u_x^2} + b(u_x)\sqrt{1 + u_x^2},$$

with $a(p),b(p)$ derived from $A,B$ via

$$a(p) = A\left(\frac{(-p,1)}{\sqrt{1+p^2}}\right), \qquad b(p) = B\left(\frac{(-p,1)}{\sqrt{1+p^2}}\right),$$

as in (Yuan et al., 2020).

More generally, for hypersurfaces $M_t \subset \mathbb{R}^{n+1}$, the normal velocity often takes the form

$V = \operatorname{div}_{M_t} D\gamma(\nu),$

with $\gamma$ a smooth, positive, one-homogeneous anisotropy function and $D\gamma$ its gradient, or via the Cahn–Hoffman field (Cui et al., 25 Oct 2025, Mercier et al., 2016, Bellettini et al., 2023).

2. Analytic Properties: Existence, Regularity, and Barriers

Well-posedness of anisotropic curvature flows depends on the structure of the anisotropy. For smooth, elliptic (uniformly convex) anisotropies, short-time existence and uniqueness of classical (smooth) solutions for curves, hypersurfaces, and networks are established using quasilinear parabolic PDE theory (Mercier et al., 2016, Kroener et al., 2020, 2310.22136). The standard approach requires showing uniform interior and boundary gradient bounds, with comparison principles and maximum-principle arguments as key tools (Yuan et al., 2020, Cui et al., 25 Oct 2025).

Global existence and non-blow-up rely on constructing explicit time-dependent barriers. For instance, in planar band domains, “cup-like” traveling wave solutions provide upper and lower barriers, leading to global bounds such as

$ct-C_1 \leq u(x,t) \leq ct+C_2$

for positive constants $c$, $C_1$, $C_2$ determined by the anisotropy and the problem geometry (Yuan et al., 2020). For networks and multiphase structures, energy dissipation and integral curvature estimates are used, with boundary contributions controlled by generalized Herring (Young) conditions (Kroener et al., 2020, Bellettini et al., 2023).

In degenerate or crystalline cases (anisotropies with flat parts or non-smooth Wulff shapes), local existence of $C^{1,1}$ or piecewise smooth solutions follows via approximation by smooth anisotropies, and short-time evolution is governed by systems of ODEs for the edge lengths in polygonal networks (Mercier et al., 2016, Bellettini et al., 2023).

3. Asymptotic Behavior and Traveling Waves

A central theme is the asymptotic shape selection under anisotropic curvature flow. In fully anisotropic band or half-space problems, under symmetric hypotheses and suitable boundary laws, any solution converges (after height normalization) to a unique traveling wave or “translator” of the form

$u(x, t) = \varphi(x) + c t,$

where $\varphi$ solves a profile ODE such as

$$-c = a(\varphi')\frac{\varphi''}{1 + (\varphi')^2} + b(\varphi')\sqrt{1 + (\varphi')^2}$$

with boundary blow-up conditions $\varphi'(\pm 1) \rightarrow \pm \infty$ (Yuan et al., 2020).

For bounded strictly convex domains with Neumann, Dirichlet, or contact-angle conditions, solutions converge to translators determined by solutions of the corresponding elliptic eigenvalue problem, with speed $\lambda$ and boundary conditions matched to the anisotropy and domain geometry (Cui et al., 25 Oct 2025, Cui et al., 25 Oct 2025).

In planar and multiphase cases, convergence to union of Wulff shapes (critical points of the anisotropic perimeter functional) is established in area-preserving flows, with explicit exponential convergence rates and quantification via Lojasiewicz-type spectral gaps (Kim et al., 2024).

4. Role and Impact of Anisotropy

The directional dependence encoded in $A$, $B$, or more generally $\gamma$ or $F$ in higher dimensions, fundamentally alters both the qualitative and quantitative evolution. Key effects include:

• Speed selection and morphology: The unique traveling-wave/translator speed $c$ depends nontrivially on $a(p)$ and $b(p)$ via the profile ODE. The cup-like shape with infinite derivatives at the boundary is a result of the specific anisotropy (Yuan et al., 2020).
• Convexity preservation: For smooth, strictly convex initial data, anisotropy preserves convexity under sufficient regularity and structural conditions (Cui et al., 25 Oct 2025, Cui et al., 25 Oct 2025). In capillary or volume-preserving settings, convexity and star-shapedness are similarly preserved (Ding et al., 2024, Andrews et al., 2021).
• Regularity and degeneracy: Strong anisotropy or non-smoothness leads to loss of uniform parabolicity, with estimates depending on “tangential” vs. “normal” directions relative to the interface (Cui et al., 25 Oct 2025).
• Singularities and crystalline flow: For crystalline anisotropy (polygonal Wulff shapes), the flow reduces to finitely many ODEs, with singularity formation (edges vanishing or curvature blow-up) governed by network configuration and angle conditions (Mercier et al., 2016, Kroener et al., 2020, Bellettini et al., 2023).

5. Boundary Value Problems and Contact Angle Effects

Anisotropic curvature flows with boundary conditions arise in modeling droplets, capillarity, and crystalline facet growth. Two canonical classes of boundary conditions are:

• Contact-angle (capillary) conditions: Prescribing the angle between the surface normal and an external (usually vertical or substrate) direction, with the law

$D_N u = -\sqrt{1 + |Du|^2} \cos\theta,$

which, in the anisotropic setting, corresponds to Young’s law generalized to the anisotropy (Cui et al., 25 Oct 2025, Ding et al., 2024).

• Neumann or Dirichlet conditions: Prescribing fluxes or heights along the boundary, allowing Dirichlet-type or more general oblique derivative problems (Cui et al., 25 Oct 2025).

Global a priori gradient bounds are secured via maximum principle applied to auxiliary functions involving the gradient, domain-defining function, and the anisotropic structure. Asymptotic convergence to translating profiles is established by compactness and the decay of oscillation of the remainder (Cui et al., 25 Oct 2025, Cui et al., 25 Oct 2025).

6. Variational and Minimizing-Movement Formulations

Weak (variational) formulations, particularly via the minimizing-movements (De Giorgi–Almgren–Taylor–Wang) scheme, play a pivotal role in both the theory and numerics of anisotropic curvature flows (Chambolle et al., 2020, Bellettini et al., 2020, Kholmatov, 2024). The incremental step minimizes an anisotropic perimeter plus a (mobility-weighted) distance penalty, often with boundary or volume constraints and possibly forcing terms. For smooth convex anisotropy and strictly mean-convex or outward-minimizing initial data, these schemes:

• Preserve mean-convexity,
• Achieve strict convergence of time-integrated perimeters, and
• Guarantee uniqueness of the limit (“flat flow”) (Chambolle et al., 2020).

Consistency between weak (minimizing-movement) and classical PDE solutions is established when smooth solutions exist, and weak comparison principles ensure order-preservation under initial data and forcing (Kholmatov, 2024, Bellettini et al., 2020).

7. Applications, Further Directions, and Key Open Problems

Anisotropic curvature flows underpin a broad spectrum of applications in materials science (grain growth, crystal facet dynamics), capillarity, shape optimization, and image processing. Flows with prescribed contact angles model droplets on inhomogeneous substrates, with boundary effects critically shaping the evolving geometry (Kholmatov, 2024, Cui et al., 25 Oct 2025, Cui et al., 25 Oct 2025). Volume-preserving variants govern relaxation to Wulff shapes, with convergence to minimizers of anisotropic isoperimetric-type inequalities and capillary Alexandrov-Fenchel inequalities (Ding et al., 2024, Andrews et al., 2021).

Outstanding challenges include:

• Long-time singularity formation and classification under strong or crystalline anisotropy,
• Weak-strong uniqueness and stability in multidimensional, multiphase settings,
• Consistency and regularity of minimization schemes for polycrystalline or non-smooth energies,
• Precise characterization of the asymptotic spectrum of stationary solutions (translators, shrinkers, Wulff-disk unions) under anisotropic flow with various geometric constraints.

The theoretical framework established in these works provides a robust analytic and variational toolkit for addressing these topics in depth (Yuan et al., 2020, Cui et al., 25 Oct 2025, Kholmatov, 2024, Chambolle et al., 2020, Mercier et al., 2016, Andrews et al., 2021).