MSC: 35C05,35D40,49L12,49N70
DOI: 10.21538/0134-4889-2021-27-3-237-245
Full text
This is a short survey of recent results obtained by the author and collaborators primarily on Hopf–Lax formulas for Hamilton–Jacobi equations and obstacle problems. The initiation of the use of quasiconvex (i.e., level convex) functions in $L^\infty$ control and differential games led to such formulas and is briefly reviewed. Dedicated to the memory of Academician A.I. Subbotin.
Keywords: Hopf–Lax, viscosity solution, Hamilton–Jacobi, quasiconvex
REFERENCES
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Received March 17, 2021
Revised May 11, 2021
Accepted May 24, 2021
Emmanuel N. Barron, Professor of Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois, 60660, USA, e-mail: ebarron@luc.edu
Cite this article as: E.N. Barron. A Survey of Hopf–Lax Formulas and Quasiconvexity in PDEs, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 237–245.
Русский
Е.Н. Баррон. Обзор формул Хопфа — Лакса и квазивыпуклость дифференциальных уравнений в частных производных
Статья представляет собой краткий обзор результатов, полученных автором и коллегами и касающихся в основном формул Хопфа — Лакса для уравнений Гамильтона — Якоби и задач с препятствием. К использованию таких формул привело начало применения квазивыпуклых функций (т. е. функций с выпуклыми множествами уровня) для управления в $L^\infty$ и дифференциальных игр, что также рассматривается в обзоре. Посвящается памяти академика А.И. Субботина.
Ключевые слова: Хопф, Лакс, вязкостное решение, Гамильтон, Якоби, квазивыпуклость
