How to integrate
$\int \ln{|\nabla u|^2}\, \mathrm{d}u$ ?
In Cartesian co-ordinates this would be $\int \ln{(u_x^2 + u_y^2)}\, \mathrm{d}u$, where $u_x \equiv \frac{\partial u}{\partial x}$. We know $\nabla^2 u = 0$, but nothing more. $u$ is a two-dimensional real-valued field; in Cartesians $u = u(x,y)$.
I do not have any idea how to treat the $x$ and $y$. I guess one could try $\mathrm{d}u = \frac{\partial u}{\partial x}\mathrm{d}x + \frac{\partial u}{\partial y}\mathrm{d}y$, but this seems to only make it worse. Is this kind of an integral even well defined?
Is the 'easier' integral (where we know $\nabla^2 u = 0$)
$\int |\nabla u|^2\, \mathrm{d}u = \frac{1}{2} \int \nabla^2 u^2\, \mathrm{d}u$
different or doable?
We can write this as $\int |\nabla u|^2\, \nabla u \cdot \mathrm{d} \textbf{r}$. Then the question is equivalently how to find $\phi$ such that $\nabla \phi = \nabla^2 u^2 \nabla u$, for use in the gradient theorem to reduce the integral to a boundary term.