2010 # 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 …

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“…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.…”

confidence: 64%

“…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.…”

confidence: 64%

“…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]).…”

confidence: 95%

“…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.,…”

confidence: 99%

“…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.…”

confidence: 99%