On a class of special Euler-Lagrange equations

TL;DR

This paper investigates special solutions to a class of Euler-Lagrange equations derived from an energy functional involving the determinant of the gradient, revealing conditions for solutions to be energy minimizers and exploring various solution types in 2D.

Contribution

It characterizes when weak solutions have a constant derivative of the energy function and explores properties of solutions, including homeomorphisms and radial solutions, in 2D cases.

Findings

Weak solutions with $f'( ext{det} Du)$ constant are energy minimizers.

Existence of weak solutions where $f'( ext{det} Du)$ is not constant.

Results on homeomorphism solutions and properties in 2-D cases.

Abstract

We make some remarks on the Euler-Lagrange equation of energy functional $I(u)=\int_\Omega f(\det Du)\,dx,$ 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-quasimonotonicty, radial solutions, and some special properties and questions in the 2-D cases.