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Rectifiability of Line Defects in Liquid Crystals with Variable Degree of Orientation

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Abstract

In [2], Hardt, Lin and the author proved that the defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Hölder continuous curves with finitely many crossings. In this article, we show that each Hölder continuous curve in the defect set is of finite length. Hence, locally, the defect set is rectifiable. For the most part, the proof closely follows the work of De Lellis et al. (Rectifiability and upper minkowski bounds for singularities of harmonic q-valued maps, arXiv:1612.01813, 2016) on harmonic \({\mathcal{Q}}\)-valued maps. The blow-up analysis in Alper et al. (Calc Var Partial Differ Equ 56(5):128, 2017) allows us to simplify the covering arguments in [11] and locally estimate the length of line defects in a geometric fashion.

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References

  1. Almgren F.J. Jr.: Q valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Am. Math. Soc. (N.S.) 8(2), 327–328 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alper O., Hardt R., Lin F.-H.: Defects of liquid crystals with variable degree of orientation. Calc. Var. Partial Differ. Equ. 56(5), 128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio L., Virga E.G.: A boundary value problem for nematic liquid crystals with a variable degree of orientation. Arch. Ration. Mech. Anal. 114(4), 335–347 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aronszajn N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36(9), 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  5. Azzam J., Tolsa X.: Characterization of n-rectifiability in terms of Jones’ square function: part II. Geom. Funct. Anal. 25(5), 1371–1412 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ball J.M., Zarnescu A.: Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli L.A., Lin F.-H.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21(3), 847–862 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L.A.; Lin, F.-H.: Analysis on the junctions of domain walls. Discrete Contin. Dyn. Syst. 28(3), 915–929 2010

  9. Caffarelli, L.A.; Lin, F.-H.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1–2), 5–18 2007

  10. David, G., Semmes, S.: Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, Vol. 38. American Mathematical Society, Providence, 1993

  11. de Lellis, C., Marchese, A., Spadaro, E., Valtorta, D.: Rectifiability and upper minkowski bounds for singularities of harmonic q-valued maps, arXiv:1612.01813 2016

  12. De Lellis C., Spadaro E.: Regularity of area minimizing currents III: blow-up. Ann. Math. 183(2), 577–617 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ericksen J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Focardi, M., Spadaro, E.: On the measure and the structure of the free boundary of the lower dimensional obstacle problem, arXiv:1703.00678 2017

  15. Hardt R., Kinderlehrer D., Lin F.-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Hardt R., Lin F.-H.: The singular set of an energy minimizing map from B 4 to S 2. Manuscr. Math. 69(3), 275–289 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. HardtR., Lin F.-H.: Harmonic maps into round cones and singularities of nematic liquid crystals. Math. Z. 213(4), 575–593 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jones P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102(1), 1–15 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Krummel, B., Wickramasekera, N.: Fine properties of branch point singularities: two-valued harmonic functions, arXiv:1311.0923 2013

  20. Lin F.-H.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42(6), 789–814 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin F.-H.: On nematic liquid crystals with variable degree of orientation. Commun. Pure Appl. Math. 44(4), 453–468 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin F.-H., Poon C.-C.: On Ericksen’s model for liquid crystals. J. Geom. Anal. 4(3), 379–392 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Maddocks, J.H.; A model for disclinations in nematic liquid crystals, Theory and Applications of Liquid Crystals Minneapolis, Minn.: IMA Vol. Math. Appl., Vol. 5. Springer, New York, 1987, 255–269 1985

  24. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Vol. 44. Cambridge University Press, Cambridge, 1995

  25. Naber A., Valtorta D.: Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Math. (2) 185(1), 131–227 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Reifenberg E.R.: Solution of the Plateau Problem for m-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schoen R., Uhlenbeck K.: A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Simon L.: Rectifiability of the singular set of energy minimizing maps. Calc. Var. Partial Differ. Equ. 3(1), 1–65 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Based on Lecture Notes by Norbert Hungerbühler. Birkhäuser Verlag, Basel, 1996

  30. Weiss G.S.: Partial regularity for weak solutions of an elliptic free boundary problem. Commun. Partial Differ. Equ. 23(3–4), 439–455 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA

    Onur Alper

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  1. Onur Alper
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Correspondence to Onur Alper.

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Communicated by C. De Lellis

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Alper, O. Rectifiability of Line Defects in Liquid Crystals with Variable Degree of Orientation. Arch Rational Mech Anal 228, 309–339 (2018). https://doi.org/10.1007/s00205-017-1193-1

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