Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

Abstract

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hölder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs.

Area minimizing discs in locally non-compact metric spaces

“…The theorem furthermore holds with the Reshetnyak energy E p + (u) replaced by the Korevaar-Schoen Dirichlet energy E p (u) defined in [14]. The theorem generalizes for example [16,Theorem 2.3] and [14,Theorem 2.2]. For regularity results for solutions of Dirichlet's problem in the metric space setting we refer for example to [14] and [16] and the references therein.…”

Regularity of quasi-n-harmonic mappings into NPC spaces

“…Based on the recent solution of Plateau's problem in proper metric spaces [33], Lytchak and Wenger [34] considered the interior regularity of quasi-harmonic mappings from twodimensional Euclidean domains to proper metric spaces. They proved that each quasiharmonic mapping u : Ω → X from a planar Euclidean domain to a large class of proper metric spaces has a locally Hölder continuous representative.…”

“…Motivated by the above work of Lytchak and Wenger [34], and also by the recent development of harmonic mappings in singular metric spaces, in this short note, we study interior and boundary regularity of quasi-n-harmonic mappings from Euclidean domains to NPC spaces.…”

“…However, our proof of Corollary 1.3 is more elementary and simpler, comparing with the more general proof there. On the other hand, our proof relies heavily on the special structure of NPC spaces and hence seems hard to be extended to the more general setting as considered in [34]. As to quasiharmonic mappings on higher dimensional Euclidean domains (n ≥ 3), there is no hope to derive Hölder continuity in this respect, even for quasiharmonic mappings into Euclidean domains (in this case, quasiharmonic mappings are also named quasiminima).…”

Spaces with almost Euclidean Dehn function

“…Define a metric space Y n by Y n := X n × D n , where we equip Y n with the Euclidean product metric, again denoted by d n . Notice that Y n is proper and geodesic and satisfies δ Y n (r) ≤ C ′ r 2 for all r ∈ (0, r 0 ), where C ′ only depends on C, see [23,Lemma 3.2]. View X n as a subset of Y n by identifying X n with X n × {0}.…”