Recently, Dr. Zhang Zhi, a young faculty member from the College of Mathematics and Statistics at our university, together with his collaborators, published a research paper titled “Extremal norms of potentials from fixed eigenvalues or eigenvalues ratio for Camassa-Holm equations” in Advances in Mathematics, a leading international mathematics journal. Founded in 1961, the journal is dedicated to publishing groundbreaking and significant results across all areas of pure mathematics. It is classified as a T1 journal in the Excellent Journal Directory of the Chinese Mathematical Society and is widely recognized as one of the most prestigious mathematics journals internationally, commanding a high academic reputation.
This paper investigates an inverse spectral problem for a class of second-order ordinary differential equations, focusing on the optimization of the Lebesgue norm extremum of the potential function under given eigenvalues or eigenvalue ratios. This equation corresponds to the spectral problem of the Lax pair for the Camassa-Holm equation governing shallow water waves. Research in this area has been highly active in recent years. Departing from existing approaches in the literature, this paper develops an analytical method rooted in the theory of measure differential equations to address such problems. Unlike previous studies, this approach does not require assumptions such as symmetry or convexity of the potential function. The reviewers highly praised the work, stating: “The technical level is first-class and the results are of interest to several active research groups.”
This study builds upon the preliminary work of Zhang Zhi and his collaborators. Previously, they published a paper titled “Minimizations of positive periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems” in Advances in Mathematics, which laid an important theoretical foundation for the current research.
Original link to this article: https://doi.org/10.1016/j.aim.2025.110766
