[Paper Review] Effective fronts of polytope shapes

Wenjia Jing, Hung V. Tran+1 more|arXiv (Cornell University)|Sep 24, 2019

Advanced Mathematical Modeling in Engineering10 references4 citations

TL;DR

This paper establishes that for dimensions $ n \geq 3 $, all centrally symmetric polytopes with rational coordinates and nonempty interior are admissible as effective fronts in periodic homogenization of first-order Hamilton-Jacobi equations. Using PDE-based methods, the authors provide a simple proof of this inverse shape theorem and derive the optimal convergence rate for the homogenization process when the effective Hamiltonian arises from such polytope-shaped limits.

ABSTRACT

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that for $n \geq 3$, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established in [1,10] in the form of stable norms as an extension of Hedlund's classical result [7]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

Motivation & Objective

• To determine which convex sets can arise as effective fronts in periodic homogenization of first-order front propagation equations.
• To establish an inverse shape theorem identifying admissible limiting shapes for the reachable set in oscillatory media.
• To provide a PDE-based proof of admissibility for centrally symmetric polytopes with rational vertices and nonempty interior in dimensions $ n \geq 3 $.
• To derive the optimal convergence rate for the homogenization process when the effective Hamiltonian corresponds to such polytope-shaped limits.

Proposed method

• The authors analyze the periodic homogenization of Hamilton-Jacobi equations with oscillatory Hamiltonians of the form $ u_t + a(x/\varepsilon)|Du| = 0 $, where $ a $ is $ \mathbb{Z}^n $-periodic and positive.
• They use the cell problem $ a(y)|p + Dv_p(y)| = \overline{H}(p) $ to define the effective Hamiltonian $ \overline{H} $, which is shown to be the support function of the limiting reachable set.
• The reachable set $ \mathcal{R}_t(x)/t $ is studied in the limit $ t \to \infty $, and its convergence to a compact convex set $ D $ is established via the Hausdorff metric.
• A constructive path-approximation argument is used to show that any point in the interior of a centrally symmetric rational polytope can be approximated by rescaled reachable sets, leveraging periodicity and path concatenation across rational directions.
• The proof of optimal convergence rate relies on estimating the distance between the rescaled reachable set and the limit shape using geometric control and covering arguments over a finite number of directions.
• The convergence is shown to hold locally uniformly in space and time, with the rate quantified in terms of the polytope’s rational structure and the oscillation bounds of $ a $.

Experimental results

Research questions

• RQ1 Which compact convex sets can arise as the large-time limit of the reachable set in a periodic front propagation model?
• RQ2 Can every centrally symmetric polytope with rational vertices and nonempty interior be realized as the effective front for some $ \mathbb{Z}^n $-periodic velocity function $ a $?
• RQ3 What is the optimal rate of convergence for the homogenization of Hamilton-Jacobi equations when the effective Hamiltonian is generated by such polytope-shaped limits?
• RQ4 How does the structure of the effective front (e.g., rational polytope) influence the convergence rate of the homogenization process?

Key findings

• For $ n \geq 3 $, every centrally symmetric polytope with rational coordinates and nonempty interior is admissible as the effective front in the homogenization of first-order front propagation equations.
• The authors provide a new, PDE-based proof of this admissibility result, distinct from prior approaches using stable norms or dynamical systems.
• The optimal convergence rate for the homogenization process is derived and shown to be sharp when the effective Hamiltonian corresponds to a centrally symmetric rational polytope.
• The convergence rate is quantified in terms of the polytope’s rational structure and the oscillation bounds of the coefficient $ a $, with explicit dependence on $ \varepsilon $.
• The proof relies on constructing paths that approximate the vertices of the target polytope by concatenating trajectories over rational directions, using periodicity to ensure admissibility.
• The result confirms that the effective Hamiltonian is the support function of the limiting reachable set, and this link is used to characterize the shape of $ D $ via $ \overline{H}(p) = \sup_{q \in D} p \cdot q $.