An introduction to the vectorial Calculus of Variations in $\textbf{L}^\infty$ AND Aronsson PDE systems

Abstract

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.