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# On a class of special Euler-Lagrange equations

Dec 2022

We make some remarks on the Euler-Lagrange equation of energy functional$I(u)=\int_\Omega f(\det Du)\,dx,$ where $f\in C^1(\R).$ For certain weaksolutions $u$ we show that the function $f'(\det Du)$ must be a constant overthe domain $\Omega$ and thus, when $f$ is convex, all such solutions are anenergy minimizer of $I(u).$ However, other weak solutions exist such that$f'(\det Du)$ is not constant on $\Omega.$ We also prove some resultsconcerning the homeomorphism solutions, non-quasimonotonicty, radial solutions,and some special properties and questions in the 2-D cases.

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