A Regularity Theory for Fractional Harmonic Maps

Abstract

Our purpose is to define, and develop a regularity theory for, Intrinsic MinimisingFractional Harmonic Maps from Euclidean space into smooth compact Riemannianmanifolds for fractional powers strictly between zero and one. Our aims aremotivated by the theory for Intrinsic Semi-Harmonic Maps, corresponding to thepower one-half, developed by Moser [34].Our definition and methodology are based on an extension method used forthe analysis of real valued fractional harmonic functions. We define and deriveregularity properties of Fractional Harmonic Maps by regarding their domain aspart of the boundary of a half-space, equipped with a Riemannian metric whichdegenerates or becomes singular on the boundary, and considering the regularityof their extensions to this half-space.We show that Fractional Harmonic Maps, and their first order derivatives, arelocally Hölder continuous away from a set with Hausdorff dimension depending onthe dimension of the domain and the fractional power in question. We achieve thisby establishing the corresponding partial regularity of extensions of FractionalHarmonic Maps which minimise the Dirichlet energy on the half-space.To prove local Hölder continuity, we develop several results in the spirit ofthe regularity theory for harmonic maps. We combine a monotonicity formulawith the construction of comparison maps, scaling in the Poincaré inequality andresults from the theory of harmonic maps, to prove energy decay sufficient forthe application of a modified decay lemma of Morrey.Using the Hölder continuity of minimisers, we prove a bound for the essentialsupremum of their gradient. Then we consider the derivatives in directionstangential to the boundary of the half-space; we establish the existence of theirgradients using difference quotients. A Caccioppoli-type inequality and scaling inthe Poincaré inequality then imply decay estimates sufficient for the applicationof the modified decay lemma to these derivatives.

Date of Award	25 Jan 2017
Original language	English
Awarding Institution	University of Bath
Supervisor	Roger Moser (Supervisor)