A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers
Abstract
We establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower semicontinuous in the variable p. Without the lower-semicontinuity in the variable p, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding L∞-functional. These improve or extend several known results in the literature.
Main Authors
Format
Articles
Research article
Published
2023
Series
Subjects
Publication in research information system
Publisher
Springer
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-202306083624Käytä tätä linkitykseen.
Review status
Peer reviewed
ISSN
0944-2669
DOI
https://doi.org/10.1007/s00526-023-02484-9
Language
English
Published in
Calculus of Variations and Partial Differential Equations
Citation
- Liu, J., & Zhou, Y. (2023). A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers. Calculus of Variations and Partial Differential Equations, 62(5), Article 144. https://doi.org/10.1007/s00526-023-02484-9
License
Funder(s)
Research Council of Finland
Funding program(s)
Research costs of Academy Research Fellow, AoF
Akatemiatutkijan tutkimuskulut, SA
Additional information about funding
The first author is supported by the Academy of Finland via the projects: Quantitative rectifiability in Euclidean and non-Euclidean spaces, Grant No. 314172, and Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, Grant No. 328846. The second author is supported by the National Natural Science Foundation of China (No. 12025102 & No. 11871088) and by the Fundamental Research Funds for the Central Universities. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Open Access funding provided by University of Jyväskylä (JYU).
Copyright© 2023 the Authors