A regularity theory for Fractional Harmonic Maps

• Main Author: Roberts, James
• Other Authors: Moser, Roger
• Published: University of Bath 2017
• Subjects: 510
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.760897

Our purpose is to define, and develop a regularity theory for, Intrinsic Minimising Fractional Harmonic Maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Our aims are motivated by the theory for Intrinsic Semi-Harmonic Maps, corresponding to the power one-half, developed by Moser. Our definition and methodology are based on an extension method used for the analysis of real valued fractional harmonic functions. We define and derive regularity properties of Fractional Harmonic Maps by regarding their domain as part of the boundary of a half-space, equipped with a Riemannian metric which degenerates or becomes singular on the boundary, and considering the regularity of their extensions to this half-space. We show that Fractional Harmonic Maps, and their first order derivatives, are locally Hölder continuous away from a set with Hausdorff dimension depending on the dimension of the domain and the fractional power in question. We achieve this by establishing the corresponding partial regularity of extensions of Fractional Harmonic Maps which minimise the Dirichlet energy on the half-space. To prove local Hölder continuity, we develop several results in the spirit of the regularity theory for harmonic maps. We combine a monotonicity formula with the construction of comparison maps, scaling in the Poincaré inequality and results from the theory of harmonic maps, to prove energy decay sufficient for the application of a modified decay lemma of Morrey. Using the Hölder continuity of minimisers, we prove a bound for the essential supremum of their gradient. Then we consider the derivatives in directions tangential to the boundary of the half-space; we establish the existence of their gradients using difference quotients. A Caccioppoli-type inequality and scaling in the Poincaré inequality then imply decay estimates sufficient for the application of the modified decay lemma to these derivatives.