AI 中文总结
本文基于第二作者的课程讲义，介绍L^∞向量变分法及其相关Aronsson偏微分方程组，说明其标量与向量理论的起源，指出该领域需新工具的重要性。
AI 中文摘要
本文是一篇说明性文章，部分基于第二作者开设的短期课程讲义，介绍变分法的子领域，该领域研究上确界泛函的向量变分问题，重点关注作为极值条件产生的相关偏微分方程组。标量理论历史较短，最早由G. Aronsson在20世纪60年代的研究中提出；向量情况则更新，最早由第二作者在21世纪初的研究中提出。L^∞变分法是众多应用领域的重要研究方向，但存在严重困难，需要不同于积分泛函经典情况及欧拉-拉格朗日方程所需的新工具。
英文摘要
In this expository article, which is partially based on the lecture notes of a short course delivered by the second appearing author, we introduce the subfield of the Calculus of Variations that is concerned with the study of vectorial variational problems for supremal functionals, with emphasis on the associated PDE systems arising as extremality conditions. The scalar theory has a rather short history, first arising in the work of G. Aronsson in the 1960s. The vectorial case is even more recent and first arose in work of the second appearing author in the early 2010s. The Calculus of Variations in $L^\infty$ is an all-important field for numerous applications, but it presents serious difficulties and requires new tools, different to those required for the classical case of integral functionals and Euler-Lagrange equations.
Comments21 pages, 12 figures, Invited paper for the conference proceedings of the Panhellenic Conference in Mathematical Analysis PCMA 2025, held in Athens (Greece) 18-20 December 2025
