Let $\;f:\mathbb R^n\rightarrow \mathbb R^m\;$ and $\;G:\mathbb R^m \rightarrow \mathbb R_{+}\;$ (NOTE: $\;n\;$ is not necessary equal to $\;m\;$). Assume the functional:
$\;I_{\mathbb R^n} (f) = \int_{\mathbb R^n} \frac{1}{2} {\vert \nabla f \vert}^2 + G(f) \;dx\;$ where $\;\nabla f=(\frac{\partial f_i}{\partial x_j})_{1\le i \le m,1\le j \le n}\;$ and $\;\vert \cdot \vert\;$is the Euclidean norm of the matrix.
Prove the Euler-Lagrange equation of the above functional is given by the system : $\;\Delta f -G_f(f)=0\;$ where $\;G_f(f)=(\frac{\partial G}{\partial f_1}, \dots, \frac{\partial G}{\partial f_m})^{T}\;$
My attempt:
I searched on my Evans PDE's book , where I found this:
The formula $\;(16)\;$, I believe is the solution to my problem. However I have trouble applying it here because I don't know what exactly is the $\;\frac{1}{2} {\vert \nabla f \vert}^2\;$.
I understand that $\;p_i^{k}\;$ from the book stands for $\;\frac{\partial f_k}{\partial x_i}\;$ but I'm a bit unsure how this appears in $\;\frac{1}{2} {\vert \nabla f \vert}^2\;$. How can I compute $\;L_{p_i^{k}}\;$ if I don't know what $\;\frac{1}{2} {\vert \nabla f \vert}^2\;$ looks like?
I would really appreciate any help because I've been stuck here!
Thanks in advance