Rectifiability and upper Minkowski bounds for singularities of harmonic $Q$-valued maps

Abstract

In this article we prove that the singular set of Dirichlet-minimizing Q-valued functions is countably (m − 2)-rectifiable and we give upper bounds for the (m − 2)-dimensional Minkowski content of the set of singular points with multiplicity Q.2010 Mathematics Subject Classification. 49Q20, 53A10, 49N60.

On the Singular Set of Free Interface in an Optimal Partition Problem

“…In Section 4 we prove the generalization of a crucial frequency pinching estimate that originated in [20] to the setting of functional (1.6).…”

“…In [34,35] Naber and Valtorta introduced several powerful techniques for studying the size and structure of the singular set of energy-minimizing or stationary maps into Riemannian manifolds. These techniques were extended by De Lellis et al in [20] to the context of energy-minimizing harmonic Q-valued functions. As motivated by the Ericksen model in liquid crystal theory, the singular set of energy-minimizing maps into cones over the real projective plane was studied in [3] by combining the approach of [20] with the blowup analysis in [4].…”

“…These techniques were extended by De Lellis et al in [20] to the context of energy-minimizing harmonic Q-valued functions. As motivated by the Ericksen model in liquid crystal theory, the singular set of energy-minimizing maps into cones over the real projective plane was studied in [3] by combining the approach of [20] with the blowup analysis in [4]. In this article, we adapt these techniques to the context of maps that are stationary points of the functional (1.6) under the variations (1.7).…”

“…The analysis in both [20] and [3] essentially concerns the preimage of the vertex of a conical target, while the specific features differ due to different properties of respective targets and blowup maps. Likewise, Focardi and Spadaro adapted the techniques of [20] to thin obstacle problems in [24,25], where they study the free boundary in its entirety. In contrast, we do not consider the preimage of 0 P N as a whole in this article.…”

“…In Section 5 we prove a pointwise bound on the Jones' 2 -number, which controls the deviation from flatness for a set in the euclidean space, in the spirit of [20,34,35].…”