Hier einfach mal der Teil der nicht läuft:
\begin{equation}
\begin{aligned}
0 &\overset{(6.5)(6.7)}{\geq } \frac{\Delta \omega (x) -\Delta \omega (y)}{\overline{\omega } } +\mu_0 m
\\ &\overset{(6.3)}{=} -(\lambda_1 -\lambda_0 )(\frac{\omega (x) -2\langle \bigtriangledown \log \phi_0 ,\bigtriangledown \omega (x) \rangle}{\overline{\omega } } - \frac{\omega (y) -2\langle \bigtriangledown \log \phi_0 ,\bigtriangledown \omega (y) \rangle}{\overline{\omega } } ) +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 ) \underbrace{\frac{\omega (x)-\omega (y)}{\overline{\omega}}}_{Q(x.y)} + \frac{2(\langle -\bigtriangledown \log \phi_0 ,\bigtriangledown \omega (x) \rangle -\langle -\bigtriangledown \log \phi_0 ,\bigtriangledown \omega (y) \rangle ) }{\overline{\omega } } +\mu_0 m
\\ \text{da wir bei} \quad (x_0 ,y_0 )\quad \text{sind ist} \quad Q(x,y)=m
\\ &\overset{(6.4}{=} -(\lambda_1 -\lambda_0 )m+2\bigtriangledown \omega (x) \frac{-\bigtriangledown \log \phi_0 (x) + \bigtriangledown \log \phi_0 (y)}{\overline{\omega } } +\mu_0 m
\\ &\overset{(6.4)}{=} -(\lambda_1 -\lambda_0 )m + 2(-\frac{m}{2} \overline{\omega } ' \cdot \frac{y-x}{|y-x|} )\cdot \frac{-\bigtriangledown \log \phi_0 (x) + \bigtriangledown \log \phi_0 (y)}{\overline{\omega } } +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 )m +m \overline{\omega} ' \frac{-(\bigtriangledown \log \phi_0 (y) - \bigtriangledown \log \phi_0 (x))\cdot \frac{y-x}{|y-x|}}{\overline{\omega } } +\mu_0 m
\\ &\overset{(2.1)}{\geq } -(\lambda_1 -\lambda_0 )m +m\overline{\omega } ' \cdot 2\frac{\pi}{D} \cdot \frac{1}{\overline{\omega } } \cdot\tan (\frac{\pi |y-x}{2D} ) +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 )m +m\cdot \frac{\pi}{D} \cos (\frac{\pi |y-x|}{2D} )\cdot 2\frac{\pi}{D} \cdot \frac{1}{\sin (\frac{\pi |y-x|}{2D} )} \cdot \tan (\frac{\pi |y-x|}{2D} ) +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 )m +2m \frac{\pi^2}{D^2} \cdot \frac{\cos (\frac{\pi |y-x|}{2D} )}{\sin (\frac{\pi |y-x|}{2D} )} \cdot \tan (\frac{\pi |y-x|}{2D} ) +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 )m +2m \frac{\pi^2}{D^2} \cdot \tan^{-1} (\frac{\pi |y-x|}{2D} )\cdot \tan (\frac{\pi |y-x|}{2D} ) +\mu_0 m
\\ &= -(\lambda_1 -\lambda_0 )m +2m \frac{\pi^2}{D^2} +\mu_0 m
\end{aligned}
\end{equation}
