Regularity of harmonic discs in spaces with quadratic isoperimetric inequality (Q502251)

The paper gives interesting new results about harmonic maps from a two-dimensional domain \(\Omega\subset{\mathbb R}\), bounded and Lipschitz, to a proper metric space \(X\). That space is assumed to admit a uniformly local quadratic isoperimetric inequality. This means the existence of \(l_0,C>0\) such that every closed Lipschitz curve in \(X\) of length \(\ell\leq l_0\) is the trace of a \(W^{1,2}\) map of a disk with area bounded by \(C\ell^2\). Instead of minimizing harmonic maps \(\Omega\to X\), the authors allow the more general \(M\)-quasiharmonic maps \(u\in W^{1,2}(\Omega,X)\). They are defined by the property that on every Lipschitz subdomain \(\Omega'\subset\Omega\), we have \(E^2_+(u_{|\Omega'})\leq ME^2_+(v)\) for all \(W^{1,2}\)-maps \(v:\Omega'\to X\) that coincide with \(u\) on \(\partial\Omega'\) in the sense of traces. Here \(E_+^2\) is the Reshetnyak energy. Under the assumptions made so far, it is proven that every \(M\)-quasiharmonic map is locally Hölder continuous in \(\Omega\) with some exponent \(\alpha>0\) that depends only on \(M\) and the \(C\) from the isoperimetric inequality. If the trace of \(u\) is continuous on \(\partial\Omega\), then \(u\) extends to a continuous map even up to the boundary. The boundary regularity can be improved to \(\alpha\)-Hölder if the boundary trace is Lipschitz. The proof (using methods by Morrey) relies on an ``energy filling energy'' stating that the local quadratic isoperimetric inequality implies the following. There exists \(p>2\) such that any closed Lipschitz curve \(S^1\to X\) of length \(\ell\leq l_0\) is the boundary trace of a mapping \(u\in W^{1,p}(D^2,X)\) with Reshetnyak \(p\)-energy bounded by a constant times \(\ell^p\).