A regularity theory for intrinsic minimising fractional harmonic maps

Abstract

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions. Mathematics Subject ClassificationCommunicated by M. Struwe. James Roberts was supported by an Engineering and Physical Sciences Research Council DTA Studentship for much of this research. B James Roberts

Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

“…The regularity of harmonic mappings of Riemannian manifolds with free or constrained boundary data has been increasingly analysed in recent years in view of the connection between these maps and recent definitions of fractional harmonic mappings of Riemannian manifolds, see for example [16,17,19,24]. In order to obtain (partial) regularity for fractional harmonic maps, one is led to the study of free boundary harmonic maps on domains where the Riemannian metric may degenerate or become singular along part of the boundary of the domain, depending on the fractional power in question.…”

“…In [24], the second author established partial regularity up to the boundary for minimisers v : M → N of E β with respect to a free boundary condition in the case where M is a Euclidean half-space. In this article, we will establish partial regularity for maps which are merely critical points of E β with respect to both outer and inner variations and still with a free boundary condition.…”

“…When β = 0, Theorem 1.1 is new. There are related results of the second author [24] (studying minimisers of E β ) and of Millot, Pegon and Schikorra [16] who consider a class of stationary or minimising critical points v : R m−1 ×(0, ∞) → R n of E β , where the domain is the half-space R m−1 × (0, ∞) and v satisfies the boundary constraint…”

“…The key estimate required to prove Theorem 1.1 is a Caccioppoli-type inequality, see Lemma 8.2. For minimisers, this kind of inequality is established using comparison maps, and we used this approach to prove partial regularity for minimisers of E β in [24]. When considering stationary harmonic maps, it is no longer possible to take advantage of the minimising property and a different approach is required; we will instead take advantage of compensated compactness phenomena in the Euler-Lagrange equation.…”

“…In the context where these regularity theories were developed, higher regularity follows from known estimates for continuous or Hölder continuous solutions to semi-linear elliptic equations, see, respectively, [14] and [27] for example. In [24], the second author studied minimising free boundary harmonic maps in the case where (M, g) is a half-space with the Euclidean metric. He used related ideas and some additional arguments to show smoothness in directions tangential to free boundary despite the singular factor arising from the factor d β in the energy.…”