This is problem 8.7.4 from Evans' PDE book.
Assume $\eta: \mathbb{R}^n \to \mathbb{R}$ is $C^1$. Show that $L(P,z,x) = \eta(z) \det P$ is a null Lagrangian. Here $P$ is a $n \times n$ matrix and $z \in \mathbb{R}^n$.
Proving that a function is a null Lagrangian can be done in two (equivalent) ways. The first way is to show that any smooth function $u \in C^\infty(\Omega, \mathbb{R}^n)$ solves the system of Euler-Lagrange equations
$$
-\sum_{i=1}^n \left( L_{p_i^k}(Du,u,x) \right)_{x_i} + L_{z^k}(Du,u,x) = 0 \quad \text{ in } \Omega\,, \quad k \in \{1, 2, …, n\}\,.
$$
The other way is to show that if $f,g \in C^\infty(\Omega, \mathbb{R}^n)$ and $f=g$ on $\partial \Omega$, then
$$
\int_\Omega L(Df, f, x) \, dx = \int_\Omega L(Dg, g, x)\,.
$$
My attempt:
I have tried to solve the problem by using the Euler-Lagrange equation approach. By computing the derivatives we get
\begin{align}
\sum_{i=1}^n \left( L_{p_i^k}(Du,u,x) \right)_{x_i} &= \sum_{i=1}^n \sum_{j=1}^n \eta_{z^j} (u) u^j_{x_i} (\operatorname{cof} Du)_i^k – \eta(u) \sum_{i=1}^n (\operatorname{cof} Du)_{i, x_i}^k\,.
\end{align}
Here I have used the identity $\partial_{p_i^k} \det P = (\operatorname{cof} P)_i^k$. Now in the latter term the sum of cofactors vanish by Lemma in Evans chapter 8.1. To the first term we can use the Laplace expansion $\det (Du) = \sum_{i=1}^n u^k_{x_i} (\operatorname{cof} Du)_i^k$ and we get
$$
\sum_{i=1}^n \left( L_{p_i^k}(Du,u,x) \right)_{x_i} = \eta_{z^k} \det Du + \sum_{j\neq k} \eta_{z^j}(u) \sum_{i=1}^n u^j_{x_i} (\operatorname{cof} Du)_i^k
$$
Now because $L_{z^k}(Du,u,x) = \eta_{z^k} \det Du$, the Euler-Lagrange equations are
\begin{align}
– \sum_{j\neq k} \eta_{z^j}(u) \sum_{i=1}^n u^j_{x_i} (\operatorname{cof} Du)_i^k &= 0\,, \quad k \in \{1,2, …, n\}\,.
\end{align}
I have no idea why an arbitrary smooth function should solve this equation, so I guess investigating the integral $\int L(Du,u,x)$ may be a more fruitful approach.
Best Answer
I put some model solutions online today, you can see it there (Solutions 9 exercise nr 3) https://wiki.helsinki.fi/display/mathstatKurssit/Sobolev+spaces%2C+spring+2016