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On a class of special Euler–Lagrange equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 21 September 2023, pp. 1-24
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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.
