December  2018, 38(12): 6195-6214. doi: 10.3934/dcds.2018266

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

1. 

Université Paris Diderot, Lab. J.L.Lions (CNRS UMR 7598), Paris, France

2. 

Johns Hopkins University, Department of Mathematics, Baltimore, USA

3. 

Columbia University, Department of Mathematics, New York, USA

Dedicated to Rafael de la Llave on the occasion of his 60th birthday with admiration and friendship

Received  October 2017 Revised  April 2018 Published  September 2018

This article addresses the regularity issue for minimizing fractional harmonic maps of order s∈(0, 1/2) from an interval into a smooth manifold. Hölder continuity away from a locally finite set is established for a general target. If the target is the standard sphere, then Hölder continuity holds everywhere.

Citation: Vincent Millot, Yannick Sire, Hui Yu. Minimizing fractional harmonic maps on the real line in the supercritical regime. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6195-6214. doi: 10.3934/dcds.2018266
References:
[1]

F. Bethuel, On the singular set of stationary harmonic maps, Manuscr. Math., 78 (1993), 417-443. doi: 10.1007/BF02599324. Google Scholar

[2]

L. A. CaffarelliJ. M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331. Google Scholar

[3]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[4]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190. doi: 10.2140/apde.2011.4.149. Google Scholar

[5]

F. Da Lio and T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., 227 (2011), 1300-1348. doi: 10.1016/j.aim.2011.03.011. Google Scholar

[6]

F. Duzaar and K. Steffen, A partial regularity theorem for harmonic maps at a free boundary, Asymptotic Anal., 2 (1989), 299-343. Google Scholar

[7]

L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Ration. Mech. Anal., 116 (1991), 101-113. doi: 10.1007/BF00375587. Google Scholar

[8]

E. B. FabesC. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218. Google Scholar

[9]

R. Hardt and F. H. Lin, Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math., 15 (1987), 555-588. doi: 10.1002/cpa.3160400503. Google Scholar

[10]

R. Hardt and F. H. Lin, Partially constrained boundary conditions with energy minimizing mappings, Commun. Pure Appl. Math., 42 (1989), 309-334. doi: 10.1002/cpa.3160420306. Google Scholar

[11]

F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris S. I Math., 312 (1991), 591-596. Google Scholar

[12]

T. Horiuchi, The imbedding theorems for weighted Sobolev spaces, J. Math. Kyoto Univ., 29 (1989), 365-403. doi: 10.1215/kjm/1250520216. Google Scholar

[13]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, An Introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics 135, Cambridge University Press, 2012. doi: 10.1017/CBO9781139108133. Google Scholar

[14]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Rational Mech. Anal., 215 (2015), 125-210. doi: 10.1007/s00205-014-0776-3. Google Scholar

[15]

V. Millot, Y. Sire and K. Wang, Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces, Archive for Rational Mechanics and Analysis, (2018), 1-8, arXiv: 1610.07194). doi: 10.1007/s00205-018-1296-3. Google Scholar

[16]

R. Moser, Intrinsic semiharmonic maps, J. Geom. Anal., 21 (2011), 588-598. doi: 10.1007/s12220-010-9159-7. Google Scholar

[17]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differ. Geom., 17 (1982), 307-335. doi: 10.4310/jdg/1214436923. Google Scholar

[18]

R. Schoen and K. Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Invent. Math., 78 (1984), 89-100. doi: 10.1007/BF01388715. Google Scholar

[19]

L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhaüser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6. Google Scholar

show all references

References:
[1]

F. Bethuel, On the singular set of stationary harmonic maps, Manuscr. Math., 78 (1993), 417-443. doi: 10.1007/BF02599324. Google Scholar

[2]

L. A. CaffarelliJ. M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144. doi: 10.1002/cpa.20331. Google Scholar

[3]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar

[4]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190. doi: 10.2140/apde.2011.4.149. Google Scholar

[5]

F. Da Lio and T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., 227 (2011), 1300-1348. doi: 10.1016/j.aim.2011.03.011. Google Scholar

[6]

F. Duzaar and K. Steffen, A partial regularity theorem for harmonic maps at a free boundary, Asymptotic Anal., 2 (1989), 299-343. Google Scholar

[7]

L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Ration. Mech. Anal., 116 (1991), 101-113. doi: 10.1007/BF00375587. Google Scholar

[8]

E. B. FabesC. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218. Google Scholar

[9]

R. Hardt and F. H. Lin, Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math., 15 (1987), 555-588. doi: 10.1002/cpa.3160400503. Google Scholar

[10]

R. Hardt and F. H. Lin, Partially constrained boundary conditions with energy minimizing mappings, Commun. Pure Appl. Math., 42 (1989), 309-334. doi: 10.1002/cpa.3160420306. Google Scholar

[11]

F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris S. I Math., 312 (1991), 591-596. Google Scholar

[12]

T. Horiuchi, The imbedding theorems for weighted Sobolev spaces, J. Math. Kyoto Univ., 29 (1989), 365-403. doi: 10.1215/kjm/1250520216. Google Scholar

[13]

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, An Introduction to Geometric Measure Theory, Cambridge Studies in Adavanced Mathematics 135, Cambridge University Press, 2012. doi: 10.1017/CBO9781139108133. Google Scholar

[14]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Rational Mech. Anal., 215 (2015), 125-210. doi: 10.1007/s00205-014-0776-3. Google Scholar

[15]

V. Millot, Y. Sire and K. Wang, Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces, Archive for Rational Mechanics and Analysis, (2018), 1-8, arXiv: 1610.07194). doi: 10.1007/s00205-018-1296-3. Google Scholar

[16]

R. Moser, Intrinsic semiharmonic maps, J. Geom. Anal., 21 (2011), 588-598. doi: 10.1007/s12220-010-9159-7. Google Scholar

[17]

R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differ. Geom., 17 (1982), 307-335. doi: 10.4310/jdg/1214436923. Google Scholar

[18]

R. Schoen and K. Uhlenbeck, Regularity of minimizing harmonic maps into the sphere, Invent. Math., 78 (1984), 89-100. doi: 10.1007/BF01388715. Google Scholar

[19]

L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhaüser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6. Google Scholar

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