Intrinsic Semiharmonic Maps

Abstract

For maps from a domain ⊂ R m into a Riemannian manifold N , a functional coming from the norm of a fractional Sobolev space has recently been studied by Da Lio and Rivière. An intrinsically defined functional with a similar behavior also exists, and its first variation can be identified with a Dirichlet-to-Neumann map belonging to the harmonic map problem. The critical points have regularity properties analogous to harmonic maps.Keywords Harmonic map · Dirichlet-to-Neumann map · Regularity Mathematics Subject Classification (2000) 58E20 · 35J50 · 35S99 Harmonic maps between Riemannian manifolds are particularly interesting on a twodimensional domain, because in this case, the problem is invariant under conformal transformations. Using this fact as a motivation, Da Lio and Rivière [2] proposed to study a functional for maps on a one-dimensional domain with related properties. This functional is given in terms of the seminorm belonging to the homogeneous fractional Sobolev spaceḢ 1/2 (R). Such a quantity is very natural from the analytic point of view, and the resulting problem permits the use of tools from harmonic analysis such as a Littlewood-Paley decomposition. The relationship with conformal transformations becomes apparent when the domain is regarded as the boundary of a half-plane. The energy is invariant under the Möbius transformations that map the half-plane onto itself.Communicated by Alexander Isaev. R. Moser ( )

On a Fractional Ginzburg–Landau Equation and 1/2-Harmonic Maps into Spheres

“…We finally point out that the relation between fractional harmonic maps and harmonic maps with a free boundary has been previously noticed by MOSER [57] for a (non-explicit) operator slightly different from the square root Laplacian (−∆) 1 2 , and leading to slightly weaker results.…”

Minimizing fractional harmonic maps on the real line in the supercritical regime

“…This was achieved in [6,7], thus extending the famous regularity result of F. Hélein for classical harmonic maps on surfaces [11]. The notion of 1/2-harmonic maps has been then extended in [14,16] to higher dimensions, and partial regularity for minimizing or stationary 1/2-harmonic maps established (in analogy with the classical harmonic map problem [1,5,17]).…”

Minimizing 1/2-harmonic maps into spheres

“…Assuming that v ∈ L ∞ (G), one may apply the (partial) regularity results of [17,25] to derive the following theorem (see [30,Section 4] or [33]). In its statement, sing(v) denotes the so-called singular set of v (in ∂ 0 G), i.e.,…”

“…However, some partial regularity does hold for minimizing (or at least stationary) 1/2-harmonic maps. The result of [30,33] asserts that a minimizing 1/2-harmonic map u in Ω belongs to C ∞ Ω \ sing(u) where sing(u) is the singular set of u in Ω defined as sing(u) := Ω \ x ∈ Ω : u is continuous in a neighborhood of x , (1.4) which is a relatively closed subset of Ω. Moreover, dim H sing(u) n − 2 for n 3, and sing(u) is locally finite in Ω for n = 2 (the notation dim H stands for the Hausdorff dimension), see Corollary 3.7.…”

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In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of [30,33] in the case where the target manifold is the (m − 1)-dimensional sphere. For m 3, we show that minimizing 1/2-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions.

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