On a class of special Euler-Lagrange equations

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 $f^{\prime }(\det Du)$ 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 $f^{\prime }(\det Du)$ 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.

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