On a class of special Euler-Lagrange equations

2022/12/23 by Baisheng Yan, Yan, Baisheng
Mathematics · Engineering · #Advanced Differential Equations and Dynamical Systems #Stability and Controllability of Differential Equations #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2212.12481

Abstract

We make some remarks on the Euler-Lagrange equation of energy functional I(u)=∫_Ωf(det Du) dx, where f∈ C¹(\mathbb R). For certain weak solutions u we show that the function f'(det Du) must be a constant over the domain Ω 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 Ω. We also prove some results concerning the homeomorphism solutions, non-quasimonotonicty, radial solutions, and some special properties and questions in the 2-D cases.

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