A regularity result for polyharmonic maps with higher integrability
(Extrinsically) \(k\)-polyharmonic maps \(u:\Omega\to N\) from a domain \(\Omega\) in \(\mathbb R^m\) to a compact Riemannian submanifold \(N\subseteq\mathbb R^n\) are the critical points of the higher order energy \(E^k(u):=\int_\Omega|D^ku|^2\,dx\). The natural space in which the direct method gives existence minimizers of \(E^k\) (which are weakly polyharmonic) is the Sobolev space \(W^{k,2}(\Omega,N)\). In the paper, the authors prove some regularity result for weakly polyharmonic maps which holds under two assumptions. The more natural assumption is that of smallness in some presumably natural Morrey spaces. For harmonic and biharmonic maps (i.e., \(k\in\{1,2\}\)), a monotonicity formula shows that stationary weak solutions are in these Morrey spaces, and examples show that without this smallness assumption there can be singularities. However, for \(k\geq3\), the monotonicity formula is still an open problem. The less natural assumption is \(u\in W^{k,2p}\) for some \(p>1\), which does not fit together with the existence results. Since the nonlinearities of the Euler-Lagrange equation are no longer critical in this space (at least under the Morrey space assumption), this second assumption simplifies the regularity proof considerably, the main ingredient being a suitable form of the Gagliardo-Nirenberg inequality. Even though the assumptions therefore must be considered rather strong, it should be noted that by now the paper presents the only available regularity result for polyharmonic maps in the ``supercritical'' case \(m>2k\).
scientific article; zbMATH DE number 5577695
scientific article