蒋永生：Variational analysis of the planar Lp dual Minkowski problem

【学术期刊】《Mathematische Annalen》，2022年7月。《Mathematische Annalen》（德国数学年刊）是有悠久历史的综合性数学学术期刊，以选稿严格著称，是数学界公认的国际顶级期刊，曾发表许多奠基性研究成果，有非常高的学术影响力。

【作者简介】蒋永生，中南财经政法大学统计与数学学院教授，博士生导师。主要研究方向包括应用数学、金融数学等。研究成果发表在国际专业权威期刊上，并主持过多项国家自然科学基金项目。

【主要观点】The Lp dual curvature measures were recently introduced by Lutwak et al. (Adv Math 329:85-132, 2018) to unify the classical theory of mixed volumes and the newer theory of dual mixed volumes of convex bodies. However, the associated Lp dual Minkowski problems for many important special cases remain open problems. They are analytically equivalent to a class of nonlinear problems with indices p and q. In most previous studies, conventional geometric inequalities and Aleksandrov's variational formula for convex bodies were used to study these problems.  In this paper, by using a new investigation method via directly studying the nonlinear problems characterizing the planar Lp dual Minkowski problem in Sobolev spaces, several sharp functional inequalities associated with the Lp dual curvature measures are established to generalize the classical inequalities, such as the Wirtinger's inequality and the Blaschke-Santal\'{o} inequality. Based on these new sharp functional inequalities, various existence results for the planar Lp dual Minkowski problem are obtained for integrable data and general. Compared with the uniqueness results of periodic solutions for  and  (in Dohmen and Giga, Proc Japan Acad Ser A Math Sci, 1994; Andrews, J Amer Math Soc, 2003), the non-uniqueness and multiple periodic solutions are also obtained in this paper for  and all  through variational analysis.

【论文信息介绍】该文是我校蒋永生教授与武汉理工大学王征平教授，澳大利亚Curtin大学Wu Yong Hong教授合作完成。