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

Millot, Vincent; Sire, Yannick; Yu, Hui

doi:10.3934/dcds.2018266

“…The regularity theory for extrinsic 1 2 -harmonic maps has been extended to a range of fractional powers for maps into general target manifolds N by Da Lio et al [6,7,9,35,36] etc, as well as Millot and Sire [25] for the power 1 2 . Millot-Sire and Yu have also considered the regularity of extrinsic minimising fractional harmonic maps defined on the real line for powers in (0, 1 2 ) [26]. The methods Da Lio, Rivière and Schikorra used to obtain regularity take advantage of compensation phenomena, namely higher than expected regularity which can be derived from the Euler-Lagrange equations for the energies they considered.…”

Section: Introductionmentioning

confidence: 99%

A regularity theory for intrinsic minimising fractional harmonic maps

Roberts

¹

2018

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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

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“…The regularity theory for extrinsic 1 2 -harmonic maps has been extended to a range of fractional powers for maps into general target manifolds N by Da Lio et al [6,7,9,35,36] etc, as well as Millot and Sire [25] for the power 1 2 . Millot-Sire and Yu have also considered the regularity of extrinsic minimising fractional harmonic maps defined on the real line for powers in (0, 1 2 ) [26]. The methods Da Lio, Rivière and Schikorra used to obtain regularity take advantage of compensation phenomena, namely higher than expected regularity which can be derived from the Euler-Lagrange equations for the energies they considered.…”

mentioning

confidence: 99%

The effect of the domain topology on the number of solutions of fractional Laplace problems

Figueiredo

¹

,

Bisci

²

,

Servadei

³

2018

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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.

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“…Finally, in the minimizing case and s ∈ (0, 1/2), we obtain an improvement on the size of the singular set (compared to the stationary case) from the triviality of the so-called "tangent maps" (i.e. blow-up limits), a consequence of the regularity of minimizing s-harmonic maps in one dimension proved in [34].…”

Section: Introductionmentioning

confidence: 69%