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

作者：Tran Hung V; Yu Yifeng^*

来源：Advances in Mathematics, 2018, 334: 300-321.

DOI：10.1016/j.aim.2018.06.017

• Asymptotic expansion
• Effective Hamiltonian
• Inverse problem
• Periodic homogenization
• Rigidity result
• Trigonometric polynomials

We continue studying an inverse problem in the theory of periodic homogenization of Hamilton-Jacobi equations proposed in [14]. Let V-1, V-2 is an element of C(R-n) be two given potentials which are Z(n)-periodic, and H-1, H-2 be the effective Hamiltonians associated with the Hamiltonians 1/2|p|(2)+V-1 , 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
(H) over bar (1) double left right arrow (H) over bar (2) (X) = V2 re + x(0)) for all x is an element of = R-2/Z(2),
for some c is an element of Q \ {0} and x(0) is an element of T-2. When n >= 3, the scenario is slightly more subtle, and a complete description is provided for any dimension. These resolve partially a conjecture stated in [14]. Some other related results and open problems are also discussed. 2018 Elsevier Inc.

• 出版日期2018-8-20