I've run into the question of finding the minimum (or infimum) of this integral over $C^2([0,1]^2)$, subject to the conditions that... $$\det (Du) \neq 0 \,\,\text{everywhere} \qquad \text{and} \qquad \int_{[0,1]^2} Du(x)dx = I_{2\times2} $$
Let $u: \mathbb{R}^2 \to \mathbb{R}^2$ with $u\in C^2([0,1]^2)$. Let $v \in \mathbb{R}^2\setminus \{0\}$. Find the infimum of $J$ with respect to $u$.
$$J(u) = \int_{[0,1]^2} \frac{1}{|Du(x) \cdot v|^2}dx$$
The idea is that $u$ is some deformation of the unit square. My conjecture is that the minimisers $u^*$ will be linear, and so I attempted to try and use some sort of convexity of $f(y) = \frac{1}{|y|^2}$, but I struggled to get anywhere with that strategy.
Is there a standard method for problems like this? Examples of similar solved problems would be helpful too if you have them.
Context: I'm using a continuum approximation of a system of point particles with some interaction and this is the integral which describes their total interaction energies up to some asymptotically small error term.
I believe there is no minimizer of this problem since $1/|y|^2$ is not convex on $\mathbb{R}^2\setminus \{0\}$. Consider for example $v=(1,0)$ and $u_h \in C^2([0,1]^2)$ defined by $u(x)=(x_1,h(x_1)+x_2)$ with $h:\mathbb{R}\to \mathbb{R}$ smooth and $\int_0^1 h'(x_1)\,dx_1=0$. The derivative of $u$ is given by $$Du(x)=\begin{pmatrix}1 & 0 \\ h'(x_1) & 1\end{pmatrix},$$ which satisfies the requirements. However, the energy is equal to $$J(u_h)=\int_0^1 \frac{1}{1+h'(x_1)^2}\,dx_1,$$ which can be made arbitrarily small by choosing $h$ accordingly (e.g. $h(x_1)=n\sin(2\pi x_1)$ with $n\to \infty$). This shows the infimum is zero.