Archive for Rational Mechanics and Analysis

, Volume 229, Issue 2, pp 709–788 | Cite as

Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data

  • Armin SchikorraEmail author


We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, intersect perpendicularly with a support manifold. For example, harmonic maps, or H-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Rivière, that these maps satisfy a system with an antisymmetric potential, from which one can derive the interior regularity of the solution. Avoiding a reflection argument, we show that these maps satisfy along the boundary a system of equations which also exhibits a (nonlocal) antisymmetric potential that combines information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bethuel, F.: Un résultat de régularité pour les solutions de l’équation de surfaces à courbure moyenne prescrite. C. R. Acad. Sci. Paris Sér. I Math. 314(13), 1003–1007 (1992)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blatt, S.; Reiter, Ph; Schikorra, A.: Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Am. Math. Soc. 368(9), 6391–6438 (2016)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bojarski, B.; Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106(1), 77–92 (1993)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brezis, H.; Coron, J.-M.: Multiple solutions of \(H\)-systems and Rellich’s conjecture. Commun. Pure Appl. Math. 37(2), 149–187 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brezis, H., Nirenberg, L.: Degree theory and BMO. II. Compact manifolds with boundaries. Sel. Math. (N.S.) 2(3), 309–368 (1996). With an appendix by the authors and Petru MironescuGoogle Scholar
  7. 7.
    Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31(1), 7–16 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Y.; Ding, Y.: Commutators of Littlewood-Paley operators. Sci. China Ser. A 52(11), 2493–2505 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Choné, Ph: A regularity result for critical points of conformally invariant functionals. Potential Anal. 4(3), 269–296 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72(3), 247–286 (1993)Google Scholar
  11. 11.
    Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math. (2) 103(3), 611–635 (1976)Google Scholar
  12. 12.
    Da Lio, F.: Fractional harmonic maps into manifolds in odd dimension \(n > 1\). Calc. Var. Partial Differ. Equ. 48(3–4), 421–445 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Da Lio, F.: Compactness and bubble analysis for 1/2-harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 201–224 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lio, Da: F., Rivière, T.: Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps. Adv. Math. 227(3), 1300–1348 (2011)Google Scholar
  15. 15.
    Da Lio, F.; Rivière, T.: Three-term commutator estimates and the regularity of \(\frac{1}{2}\)-harmonic maps into spheres. Anal. PDE 4(1), 149–190 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Da Lio, F., Rivière, T.: Horizontal \(\alpha \)-harmonic maps, Preprint, arXiv:1604.05461 (2016)
  17. 17.
    Da Lio, F.; Schikorra, A.: \(n/p\)-harmonic maps: regularity for the sphere case. Adv. Calc. Var. 7(1), 1–26 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33(1), 263–321 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, Vol. 105. Princeton University Press, Princeton, 1983Google Scholar
  20. 20.
    Goldstein, P., Zatorska-Goldstein, A.: Remarks on Uhlenbeck’s decomposition theorem, Preprint, arXiv:1704.03550 (2017)
  21. 21.
    Grafakos, L.: Classical Fourier Analysis, Graduate Texts in Mathematics, 3rd edn., Vol. 249. Springer, New York, 2014Google Scholar
  22. 22.
    Grüter, M.: Conformally invariant variational integrals and the removability of isolated singularities. Manuscr. Math. 47(1–3), 85–104 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Grüter, M.; Hildebrandt, S.; Nitsche, J.C.C.: Regularity for stationary surfaces of constant mean curvature with free boundaries. Acta Math. 156(1–2), 119–152 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guliyev, V.; Omarova, M.; Sawano, Y.: Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces. Banach J. Math. Anal. 9(2), 44–62 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une sphère. C. R. Acad. Sci. Paris Sér. I Math. 311(9), 519–524 (1990)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I Math. 312(8), 591–596 (1991)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, Cambridge Tracts in Mathematics, 2nd edn., Vol. 150. Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original, With a foreword by James EellsGoogle Scholar
  29. 29.
    Iwaniec, T.; Martin, G.: Riesz transforms and related singular integrals. J. Reine Angew. Math. 473, 25–57 (1996)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lamm, T.; Rivière, T.: Conservation laws for fourth order systems in four dimensions. Comm. Partial Differ. Equ. 33(1–3), 245–262 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lenzmann, E., Schikorra, A.: Sharp commutator estimates via harmonic extension, Preprint, arxiv:1609.08547 (2016)
  32. 32.
    Maalaoui, A.; Martinazzi, L.; Schikorra, A.: Blow-up behavior of a fractional Adams-Moser-Trudinger-type inequality in odd dimension. Commun. Partial Differ. Equ. 41(10), 1593–1618 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mazowiecka, K., Schikorra, A.: Fractional div-curl quantities and applications to nonlocal geometric equations, arXiv: 1703.00231 (2017)
  34. 34.
    Millot, V.; Sire, Y.: On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres. Arch. Ration. Mech. Anal. 215(1), 125–210 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Moser, R.: An \(L^p\) regularity theory for harmonic maps. Trans. Am. Math. Soc. 367(1), 1–30 (2015)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Müller, F.: On stable surfaces of prescribed mean curvature with partially free boundaries. Calc. Var. Partial Differ. Equ. 24(3), 289–308 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Müller, F.; Schikorra, A.: Boundary regularity via Uhlenbeck-Rivière decomposition. Analysis (Munich) 29(2), 199–220 (2009)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Müller, S.: Higher integrability of determinants and weak convergence in \(L^1\). J. Reine Angew. Math. 412, 20–34 (1990)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Reshetnyak, Y.G.: On the stability of conformal mappings in multidimensional spaces. Sib. Math. J. 8, 65–85 (1967)Google Scholar
  40. 40.
    Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rivière, T.: Sub-criticality of Schrödinger systems with antisymmetric potentials. J. Math. Pures Appl. (9) 95(3), 260–276 (2011)Google Scholar
  42. 42.
    Rivière, T.; Struwe, M.: Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math. 61(4), 451–463 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Scheven, C.: Partial regularity for stationary harmonic maps at a free boundary. Math. Z. 253(1), 135–157 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schikorra, A.: A remark on gauge transformations and the moving frame method. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2), 503–515 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Schikorra, A.: Interior and boundary-regularity for fractional harmonic maps on domains, unpublished. arXiv:1103.5203 (2011)
  46. 46.
    Schikorra, A.: Regularity of \(n/2\)-harmonic maps into spheres. J. Differ. Equ. 252(2), 1862–1911 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Schikorra, A.: \(\varepsilon \)-regularity for systems involving non-local, antisymmetric operators. Calc. Var. Partial Differ. Equ. 54(4), 3531–3570 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Semmes, S.: A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller. Comm. Partial Differ. Equ. 19(1–2), 277–319 (1994)CrossRefzbMATHGoogle Scholar
  49. 49.
    Sharp, B.; Topping, P.: Decay estimates for Rivière’s equation, with applications to regularity and compactness. Trans. Am. Math. Soc. 365(5), 2317–2339 (2013)CrossRefzbMATHGoogle Scholar
  50. 50.
    Sharp, B., Zhu, M.: Regularity at the free boundary for Dirac-harmonic maps from surfaces. Calc. Var. Partial Differ. Equ. 55(2), Art. 27, 30 (2016)Google Scholar
  51. 51.
    Shatah, J.: Weak solutions and development of singularities of the \({{\rm SU}}(2)\) \(\sigma \)-model. Commun. Pure Appl. Math. 41(4), 459–469 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, 1970Google Scholar
  53. 53.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43. Princeton University Press, Princeton, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, IIIGoogle Scholar
  54. 54.
    Tartar, L.: The Compensated Compactness Method Applied to Systems of Conservation Laws, Systems of Nonlinear Partial Differential Equations (Oxford, 1982), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 111. Reidel, Dordrecht, 1983, pp. 263–285Google Scholar
  55. 55.
    Tomi, F.: Ein einfacher Beweis eines Regularitässatzes für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 112, 214–218 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Torchinsky, A, Wang, S.L.: A note on the Marcinkiewicz integral. Colloq. Math. 60/61(1), 235–243 (1990)Google Scholar
  57. 57.
    Uhlenbeck, K.K.: Connections with \(L^{p}\) bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations