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

Abstract

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.

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References

  1. 1.

    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    Article  MATH  Google Scholar 

  4. 4.

    Bojarski, B.; Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Stud. Math. 106(1), 77–92 (1993)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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 Mironescu

  7. 7.

    Chanillo, S.: A note on commutators. Indiana Univ. Math. J. 31(1), 7–16 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chen, Y.; Ding, Y.: Commutators of Littlewood-Paley operators. Sci. China Ser. A 52(11), 2493–2505 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Choné, Ph: A regularity result for critical points of conformally invariant functionals. Potential Anal. 4(3), 269–296 (1995)

    MathSciNet  Article  MATH  Google 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)

  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)

  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)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33(1), 263–321 (1931)

    MathSciNet  Article  MATH  Google 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, 1983

  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, 2014

  22. 22.

    Grüter, M.: Conformally invariant variational integrals and the removability of isolated singularities. Manuscr. Math. 47(1–3), 85–104 (1984)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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 Eells

  29. 29.

    Iwaniec, T.; Martin, G.: Riesz transforms and related singular integrals. J. Reine Angew. Math. 473, 25–57 (1996)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Moser, R.: An \(L^p\) regularity theory for harmonic maps. Trans. Am. Math. Soc. 367(1), 1–30 (2015)

    ADS  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Müller, F.; Schikorra, A.: Boundary regularity via Uhlenbeck-Rivière decomposition. Analysis (Munich) 29(2), 199–220 (2009)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Müller, S.: Higher integrability of determinants and weak convergence in \(L^1\). J. Reine Angew. Math. 412, 20–34 (1990)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Rivière, T.: Sub-criticality of Schrödinger systems with antisymmetric potentials. J. Math. Pures Appl. (9) 95(3), 260–276 (2011)

  42. 42.

    Rivière, T.; Struwe, M.: Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math. 61(4), 451–463 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Scheven, C.: Partial regularity for stationary harmonic maps at a free boundary. Math. Z. 253(1), 135–157 (2006)

    MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Schikorra, A.: \(\varepsilon \)-regularity for systems involving non-local, antisymmetric operators. Calc. Var. Partial Differ. Equ. 54(4), 3531–3570 (2015)

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

    Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, 1970

  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, III

  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. 263–285

  55. 55.

    Tomi, F.: Ein einfacher Beweis eines Regularitässatzes für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 112, 214–218 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Torchinsky, A, Wang, S.L.: A note on the Marcinkiewicz integral. Colloq. Math. 60/61(1), 235–243 (1990)

  57. 57.

    Uhlenbeck, K.K.: Connections with \(L^{p}\) bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  58. 58.

    Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Armin Schikorra.

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Communicated by Fanghua Lin

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Schikorra, A. Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data. Arch Rational Mech Anal 229, 709–788 (2018). https://doi.org/10.1007/s00205-018-1226-4

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