具有$3$型周期势的有效哈密顿量的刚性结果
A rigidity result for effective Hamiltonians with $3$-mode periodic potentials
摘要：
本文继续研究文献[14]中提出的Hamilton-Jacobi方程周期均匀化理论中的一个反问题。设$V_1, V_2 \in C(\mathbb{R}^n)$是两个给定的$\mathbb{Z}^n$ -周期势，$\overline{H}_1, \overline{H}_2$分别是与哈密顿量$\frac{1}{2}|p|^2 + V_1$，$\frac{1}{2}|p|^2+V_2$相关的有效哈密顿量.本文的一个主要结果是，如果维数$n=2$和每个$V_1, V_2$恰好包含$3$互不平行的Fourier模，则对于某些$c\in \mathbb{Q} \setminus\{0\}$和$x_0 \in \mathbb{T}^2$，有$$ \overline H_1\equiv \overline H_2 \quad \iff \quad V_1(x)=V_2\left({x\over c}+x_0\right) \quad \text{ for all } x \in \mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2, $$.当使用$n\geq 3$时，场景稍微微妙一些，并且为任何维度提供了完整的描述。这部分地解决了[14]中的猜想。并讨论了其它一些相关结果和尚待解决的问题。
英文摘要：
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.
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