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

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 $${\overline{H}}_1\equiv {\overline{H}}_2⟺V_1(x)=V_2\left(\frac{x}{c}+x_0\right)\text{ for all }x\in {\mathbb{T}}^2={\mathbb{R}}^2/{\mathbb{Z}}^2,$$ for some $c\in \mathbb{Q}\setminus \{0\}$ and $x_0\in {\mathbb{T}}^2$. When $n\ge 3$, 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.

18 pages, 2 figures