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On a class of special Euler–Lagrange equations

Part of: Optimality conditions Representations of solutions Equations and systems of special type

Published online by Cambridge University Press:  21 September 2023

Baisheng Yan Open the ORCID record for Baisheng Yan [Opens in a new window]
Baisheng Yan*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (yanb@msu.edu)
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Abstract

We make some remarks on the Euler–Lagrange equation of energy functional $I(u)=\int _\Omega f(\det Du)\,{\rm d}x,$ where $f\in C^1(\mathbb {R}).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $\Omega$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $\Omega.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicity and radial solutions, and finally we prove some stability results and discuss some related questions concerning certain approximate solutions in the 2-Dimensional cases.

Keywords

Special Euler–Lagrange equationshomeomorphism solutionsradial solutions

MSC classification

Primary: 35D30: Weak solutions
Secondary: 35M30: Systems of mixed type 49K20: Problems involving partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 24
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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