Euler-Lagrange Equations

Since $E(u,v)$ is highly nonlinear, the minimisation is not trivial. For better readability we define the following abbreviations, where the use of $z$ instead of $t$ emphasises that the expression is not a temporal derivative but a difference that is sought to be minimised.

$$\begin{array}{lclp{2cm}l} I_x &:=& \partial_xI(\mathbf x+\ma... ...mathbf x+\mathbf w)-\partial_yI(\mathbf x)\mbox{.} \end{array}$$ (8)

According to the calculus of variations, a minimiser of (7) must fulfill the Euler-Lagrange equations

$$\Psi'(I_z^2+\gamma(I_{xz}^2+I_{yz}^2))\cdot (I_xI_z+\gamma(I_{xx}I_{xz}+I_{xy}I_{yz}))$$

$$\qquad \qquad\qquad\qquad \qquad \qquad\qquad - \alpha\; \left(\Psi'(\vert\nabla_{\hspace*{-0.5mm}3}u\vert^2+\vert\nabla_{\hspace*{-0.5mm}3}v\vert^2)\nabla_{\hspace*{-0.5mm}3}u\right) =0,$$

$$\Psi'(I_z^2+\gamma(I_{xz}^2+I_{yz}^2))\cdot (I_yI_z+\gamma(I_{yy}I_{yz}+I_{xy}I_{xz}))$$

$$\qquad \qquad\qquad\qquad \qquad \qquad\qquad - \alpha\; \left(\Psi'(\vert\nabla_{\hspace*{-0.5mm}3}u\vert^2+\vert\nabla_{\hspace*{-0.5mm}3}v\vert^2)\nabla_{\hspace*{-0.5mm}3}v\right) =0$$

with reflecting boundary conditions.