Let we want to compute the minimizer: $$ \min \{ \int_B \{ \det\nabla u + |\nabla u|^2\}dx: u\in \mathcal{C}^2(\bar B,\mathbb{R}^2), u|_{\partial B} (x)=x\} $$ where $B=\{x\in\mathbb{R}^2: |x|<1 \}$, $x=(x_1,x_2), u=(u^1,u^2), \ u:B\to\mathbb{R}^2$ and $\det\nabla u$ is the determinant of Jacobian.
Let us work first with the $\int_B \det\nabla u \ dx$. For the case $(x_1,x_2)\to(u^1,u^2)$ we can evaluate $$ \det\nabla u = \frac{\partial}{\partial x_1}(u^1\frac{\partial u^2}{\partial x_2})-\frac{\partial}{\partial x_2}(u^1\frac{\partial u^2}{\partial x_1}) $$ therefore let us consider the following integral: $$ \int_B \frac{\partial}{\partial x_1}(u^1\frac{\partial u^2}{\partial x_2}) \ dx $$ I have a question here:
I think that it is possible to simplify the integral above and go to the integral over boundary but I am not sure how it works with the partial derivative, could you please let me know if it is even a right starting point to solve the problem and how I can proceed if it indeed is? thank you!