Quantitative regularity for $p$-harmonic maps

Abstract

In this article, we study the regularity of minimizing and stationary p-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set S(u) = {x s.t. u is not continuous at x}, as opposed to the weaker and non quantitative Hausdorff dimension bounds currently available in literature for generic p.The main technique used in this paper is the quantitative stratification, which is based on the study of the approximate symmetries of the tangent maps of u. In this article, we generalize the study carried out in [CN13b] for minimizing 2-harmonic maps to generic p ∈ (1, ∞). Moreover, we analyze also the stationary case where the lack of compactness makes the study more complicated.In order to understand the degeneracy intrinsic in the behaviour of stationary maps, we study the defect measure naturally associated to a sequence of such maps and generalize the results obtained in [Lin99a].By using refined covering arguments, we also improve the estimates in the case of isolated singularities and obtain a definite bound on the number of singular points. This result seems to be new even for minimizing 2-harmonic maps.

Some regularity results for p-harmonic mappings between Riemannian manifolds

“…General p-harmonic mappings, 1 < p < ∞, also gained growing interest in the past decades; see for instance [6,9,10,12,13,14,15,19,22,23,31,34]. Relying on the fundamental work of Struwe [44], Fardoun and Regbaoui [9,10] developed the theory of p-harmonic mapping flow and partially extended the results of Eells and Sampson [8] to p-harmonic mappings.…”

“…The structure of singular set has gained deeper understanding more recently in [23,3,34]. As to weakly p-harmonic mappings, we would like to mention the interesting work of Fardoun and Regbaoui [11], where the authors found a small constant ǫ 0 such that if u : Ω ⊂ M → N is a weakly p-harmonic mapping with u(Ω) contained in a regular geodesic ball of radius ǫ 0 , then u ∈ C 1,α (Ω, N ) for some 0 < α < 1.…”

Regularity of quasi-n-harmonic mappings into NPC spaces

“…Besides the harmonic case (p = 2) and the intermediate case (1 < p < n), the borderline case p = n also received special attention as they often enjoy better property than general p-harmonic mappings. For instance, among other results, Hardt and Lin [16] showed that minimizing n-harmonic mappings from an n-dimensional compact Riemannian manifold into a C 2 Riemannian manifold are locally C 1,α for some 0 < α < 1; Wang [47] proved that n-harmonic mappings into Riemannian manifolds (without boundary) enjoy nice compactness properties; Mou and Yang [37] obtained that n-harmonic mappings are everywhere regular in the interior, continuous up to the boundary (of a bounded smooth domain), and have removable isolated singularities; see also [38] for a recent improvement of this result. When the target metric space is the real line R, n-harmonic functions play a particularly important role in the theory of quasiconformal mappings and quasiregular mappings; see for instance [18,23] and the various references therein.…”

Moving-centre monotonicity formulae for minimal submanifolds and related equations

“…We state the classical monotonicity formula for Euclidean balls M = B m r ⊂ R m . The statement below is found in [19], but versions were proven earlier by other authors including: Schoen-Uhlenback [22] for minimising 2-harmonic maps, Price [21] for stationary 2-harmonic maps and Hardt-Lin [16] for minimising p-harmonic maps. Equivalently, for 0 < r < t <r, we have In particular, r p−m Br |∇u| p is non-decreasing in r, and is constant if and only if u is homogenous of degree zero.…”

“…In Section 3 we consider stationary p-harmonic maps (p > 1); for the fixed-centre monotonicity in this setting one may consult [19]. The minimal submanifold case is morally the p = 1 case; in fact it is the critical case for our moving-centre monotonicity in the sense that for p > 1, there is a term of the wrong sign that cannot be fully absorbed in the naive manner.…”