A rigidity result for effective Hamiltonians with $3$-mode periodic potentials

TL;DR

This paper proves a rigidity result for effective Hamiltonians derived from 2D periodic potentials with exactly three Fourier modes, showing they determine the potential up to specific transformations, and extends results to higher dimensions.

Contribution

It establishes a precise characterization of when two potentials yield the same effective Hamiltonian in 2D with three Fourier modes, advancing the inverse homogenization problem.

Findings

Effective Hamiltonians uniquely determine potentials up to rational scaling and translation in 2D with three modes.

Partial results and descriptions are provided for higher dimensions (n ≥ 3).

The work partially resolves a conjecture in the theory of periodic homogenization.

Abstract

We continue studying an inverse problem in the theory of periodic homogenization of Hamilton-Jacobi equations proposed in [14]. Let $V_1, V_2 \in C(\mathbb{R}^n)$ be two given potentials which are $\mathbb{Z}^n$-periodic, and $\overline{H}_1, \overline{H}_2$ be the effective Hamiltonians associated with the Hamiltonians $\frac{1}{2}|p|^2 + V_1$, $\frac{1}{2}|p|^2+V_2$, respectively. A main result in this paper is that, if the dimension $n=2$ and each of $V_1, V_2$ contains exactly $3$ mutually non-parallel Fourier modes, then $$ \overline H_1\equiv \overline H_2 \quad \iff \quad V_1(x)=V_2\left({x\over c}+x_0\right) \quad \text{ for all } x \in \mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2, $$ for some $c\in \mathbb{Q} \setminus\{0\}$ and $x_0 \in \mathbb{T}^2$. When $n\geq 3$, the scenario is slightly more subtle, and a complete description is provided for any dimension. These resolve partially the conjecture stated in [14]. Some other related results and open problems are also discussed.