Polyharmonic maps: conservation laws and approximations

In the first part of the thesis we consider elliptic systems in the critical dimension $2m$ that contain a term with antisymmetric structure. An example for such a system is the $m$-polyharmonic map equation which we investigate throughout the thesis. Following the work of Rivière in the two-dimensional case, we aim to write the system in divergence-free form and establish a conservation law by using a small perturbation of Uhlenbeck's gauge fixing matrix.


In the second part we focus on the $m$-polyenergy and consider the higher order approximation $E_\varepsilon(u)=\frac{1}{2}\int_\Omega(|D^m u|^2 +\varepsilon |D^{m+1}u|^2)$, which was first introduced by Lamm in the case $m=1$. We show that critical points $u_\varepsilon:\Omega\rightarrow N^n$, $\Omega\subset \mathbb{R}^{2m}$ compact without boundary, of $E_\varepsilon$ are smooth.
Further we prove that a sequence $(u_\varepsilon)$ of critical points converges strongly to an $m$-polyharmonic map away from finitely many points as $\varepsilon\rightarrow 0$. At the points of energy concentration bubbling occurs and we perform a blow-up to show convergence to quasi-$m$-polyharmonic spheres. ... mehrTo establish the energy identity in the limit we show that no energy is lost in the neck region between bubble and $m$-polyharmonic map. For mappings into the sphere this is always true. For arbitrary target manifolds $N^n$ we need to impose an additional entropy condition.


Finally, we consider the $\varepsilon$-approximation of the Dirichlet energy and investigate whether every harmonic map occurs as a limit. The answer to this question is no and we derive a gap theorem for $\varepsilon$-harmonic maps $u_\varepsilon:S^2\rightarrow S^2$ of degree zero and $\pm1$. We show that $\varepsilon$-harmonic maps of degree zero with energy below $8\pi$ are constant and maps of degree $\pm 1$ with energy below $12\pi$ are of the form $Rx$ with $R\in O(3)$. This stands in contrast to the fact that all rational maps between two-spheres are harmonic.
Moreover, we construct non-trivial $\varepsilon$-harmonic maps of degree zero with energy $\geq8\pi$.