Ribbon complementary energy

The non convex energy for a ribbon has the form:

The relaxed energy is $W^{**}(\mathbf L)$ is the largest convex minorant of $W(\mathbf L)$. For this energy, an explicit formula is contained in the paper with RP. A key point in that formula is that the determinant for a 2x2 matrix is quadratic, and hence it nicely interacts with the quadratic energy.

The complementary energy is $W^*(\mathbf M)=\displaystyle\sup_{\mathbf L\in{\rm Sym}}(\mathbf M\cdot\mathbf L-w(\mathbf L))$.

We can gather the components of $\mathbf L$ into a vector $\mathbf l=(L_{11},L_{22},L_{21})^T$. We can also introduce the matrix

Also, we can set

Then we have

Then we have $W(\mathbf L)=w(\mathbf l)$ where

Note that since $w(\mathbf l)=+\infty$ if $\mathbf S\mathbf l\cdot\mathbf l=0$, we can also write

We note that the above maximization problem needs not have a unique solution. Let $\mathbf l^*$ be any such solution. Then,

To solve the variational problem $\eqref{eq:1001}$ we introduce the Lagrangian $\mathcal L(\mathbf m,\mathbf l,\lambda)=\mathbf l\cdot\mathbf m-\frac1 2 \mathbf D\mathbf l\cdot\mathbf l-\frac\lambda 2 \mathbf S\mathbf l\cdot\mathbf l$ . The optimality conditions are

In particular, taking the scalar product of both sides of the first equation with $\mathbf l^*$, we obtain