# Polyharmonic maps: conservation laws and approximations

Hörter, Jasmin

## Abstract (englisch):

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. ... mehr

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA) Publikationstyp Hochschulschrift Publikationsdatum 31.08.2021 Sprache Englisch Identifikator KITopen-ID: 1000136367 Verlag Karlsruher Institut für Technologie (KIT) Umfang vii, 140 S. Art der Arbeit Dissertation Fakultät Fakultät für Mathematik (MATH) Institut Institut für Analysis (IANA) Prüfungsdatum 31.03.2021 Schlagwörter Polyharmonic maps, harmonic maps, conservation laws, approximate energy functionals, geometric analysis, partial differential equations, calculus of variations Referent/Betreuer Lamm, T.
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft
KITopen Landing Page