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

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.