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}$^n)$betwogivenpotentialswhichare$$\mathbb{Z}$^n$-periodic,and$\overline{H}_1, \overline{H}_2$ be the effective Hamiltonians associated with the Hamiltonians $\frac{1}{2}\mathrm{\mid }p{\mathrm{\mid }}^2+V_1$, $\frac{1}{2}\mathrm{\mid }p{\mathrm{\mid }}^2+V_2$, respectively. A main result in this paper is that, if the dimension $n=2$ and each of $V_1, V_2$containsexactly$3$mutuallynon-parallelFouriermodes,then$$ \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, $$forsome$c\in $\mathbb{Q}$ \setminus{0}$and$x_0 \in $\mathbb{T}$^2$\mathrm{.}When$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.