中科院数学与系统科学研究院
数学研究所
学术报告
非线性分析研讨班
报告人：张泽鑫博士（江苏大学）
题 目：One-dimensional symmetry and uniqueness of solutions of some elliptic systems in R^{N-1} \times (l,r) and applications to Gross-Pitaevskii systems
报告语言：中文&英文
时 间：2022.12.15（星期四）19:00-20:00
地 点：腾讯会议 800-935-156
摘 要：In this paper, we use the moving plane method to prove one-dimensional symmetry and monotonicity of solutions with priori bounds and heteroclinic boundary conditions in one direction to a general two-coupled competing elliptic system in R^{N-1}\times(l,r) . Here -\infty<l<r<+\infty are two generalized numbers. Moreover, we obtain that the system has at most one classical solution via the sliding method. These generalize some of results of Berestycki et al. [Duke Math. J. 103 (2000), no. 3, 375--396; MR1763653] to systems. We remark that the system does not need to require a variational structure. The key for the proofs is to establish a comparison result for the system and a maximum principle for some solutions of the corresponding linearized system. As an application, we deduce a priori estimates of non-negative solutions to some two-coupled Gross-Pitaevskii systems in R^{N-1}\times(l,r)) and prove the validity of the corresponding Gibbons' conjecture. Moreover, we prove the existence and uniqueness of domain wall solutions in slab domains or half spaces. Furthermore, we derive the uniform boundedness of gradients of these solutions and analyze the asymptotic behavior when the coupling coefficients tend to infinity. As a consequence, phase separation occurs and we show that the interface of the limiting profile contains exactly one point in $(l,r)$. In addition, the asymptotic behavior of the solutions near the interfaces is also studied. This work is joined with Jun Wang and Zhitao Zhang.
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