Journal d'Analyse Mathématique

, Volume 133, Issue 1, pp 335–359 | Cite as

Extremal functions for modules of systems of measures

  • Melkana Brakalova
  • Irina Markina
  • Alexander Vasil’ev


We extend a result by Rodin, which provides an explicit method for finding the extremal function and the 2-module of a foliated family of curves in R2, to R n making use of Fuglede’s p-module of systems of measures. The extremal functions are identified and the p-module of systems of measures is computed in condensers of rather general type and in their images under homeomorphisms of certain regularity. At the beginning, we discuss and apply Rodin’s Theorem in order to obtain estimates for the conformal modules of parallelograms and ring domains in terms of directional dilatations.


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  1. [1]
    R. A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975.MATHGoogle Scholar
  2. [2]
    L. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Book Co., New York, 1973.MATHGoogle Scholar
  3. [3]
    L. Ahlfors, Lectures on Quasiconformal Mappings, second edition, American Mathematical Society, Provicence, RI, 2006.CrossRefMATHGoogle Scholar
  4. [4]
    L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), 101–129.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. D. Anderson, S. L. Qiu, M. K. Vamanamurthy, and M. Vuorinen, Generalized elliptic integrals and modular equations, Pacific J. Math., 192 (2000), 1–37.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Aikawa and M. Ohtsuka, Extremal length of vector measures, Ann. Acad. Sci. Fenn. Math., 24 (1999), 61–88.MathSciNetMATHGoogle Scholar
  7. [7]
    C. Andreian Cazacu, On the length-area dilatation, Complex. Var. Theory. Appl., 50 (2005), 765–776.MathSciNetMATHGoogle Scholar
  8. [8]
    M. Badger, Beurling’s criterion and extremal metrics for Fuglede modulus, Ann. Acad. Sci. Fenn., Math., 38 (2013), 677–689.MathSciNetMATHGoogle Scholar
  9. [9]
    M. Brakalova, On the asymptotic behavior of some conformal and quasiconformal mappings, Ph.D. Thesis, Sofia University, Sofia, 1988.MATHGoogle Scholar
  10. [10]
    M. A. Brakalova and J. A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math., 76 (1998), 67–92.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Brakalova and J. A. Jenkins, On a paper of Carleson: “On mappings conformal at the boundary”, Ann. Acad. Sci. Fenn. Math., 27 (2002), 485–490.MathSciNetMATHGoogle Scholar
  12. [12]
    M. Brakalova, Sufficient and necessary conditions for conformality.Part II. Analytic viewpoint, Ann. Acad. Sci. Fenn. Math., 35 (2010), 235–254.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Brakalova, On local stability of solutions to the Beltrami equation with degeneration, Complex Anlalysis and Applications,’13, Bulgarian Acad. Sci., Sofia, 2013, pp. 65–76.Google Scholar
  14. [14]
    M. Csörnyei, S. Hencl, and Y. Malý, Homeomorphisms in the Sobolev space W 1, n-1, J. Reine Angew. Math., 644 (2010), 221–235.MathSciNetMATHGoogle Scholar
  15. [15]
    V. Dubinin and M. Vuorinen, On conformal moduli of polygonal quadrilaterals, Israel J. Math, 171 (2009), 111–125.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    A. Ya. Dubovickiĭ, On the structure of level sets of differentiable mappings of an n-dimensional cube into a k-dimensional cube, Izv. Akad. Nauk SSSR. Ser. Mat., 21 (1957), 371–408.MathSciNetGoogle Scholar
  17. [17]
    H. Federer, Geometric Measure Theory, Springer-Verlag New York Inc., New York, 1969.MATHGoogle Scholar
  18. [18]
    B. Fuglede, Extremal length and functional completion, Acta Math., 98 (1957), 171–219.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    J. Garnett and D. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005.CrossRefMATHGoogle Scholar
  20. [20]
    F.W. Gehring, Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J., 9 (1962), 137–150.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    H. Grötzsch, Eleven papers, Ber. Verh. Sächs. Acad. Wiss., Leipzig, Math. Phys. (1928–1932).Google Scholar
  22. [22]
    V. Gutlyanskii and O. Martio, Conformality of a quasiconformal mapping at a point, J. Anal. Math., 91 (2003), 179–192.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    P. Hajlasz, Change of variables formula under minimal assumptions, Colloq. Math., 64 (1993), 93–101.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    P. Hajlasz, Sobolev mappings, co-area formula and related topics, Proceedings on Analysis and Geometry, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, pp. 227–254.MATHGoogle Scholar
  25. [25]
    H. Hakula, A. Rasila, and M. Vuorinen, On moduli of rings and quadrilaterals: algorithms and experiments, SIAM J. Sci. Comput., 33 (2011), 279–302.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    J. Hesse, A p-extremal length and p-capacity equality, Ark. Mat. 13 (1975), 131–144.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, Oxford University Press, New York, 2001.MATHGoogle Scholar
  28. [28]
    J. A. Jenkins, Univalent Functions and Conformal mapping, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958.CrossRefMATHGoogle Scholar
  29. [29]
    O. Lehto, Homeomorphisms with a given dilatation, Proceedings of the 15th Scandanavian Conference, Oslo, 1968, Springer, Berlin, 1970, pp. 58–73.MATHGoogle Scholar
  30. [30]
    O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York-Heidelberg, 1973.CrossRefMATHGoogle Scholar
  31. [31]
    J. Malý, D. Swanson, and W. P. Ziemer, The co-area formula for Sobolev mappings, Trans.Amer. Math. Soc., 355 (2003), 477–492.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    I. Markina, On coincidence of p-module of a family of curves and p-capacity on the Carnot group, Rev. Mat. Iberoamericana,, 19 (2003), 143–160.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    O. Martio and W. P. Zimmer, Luzin’s condition (N) and mappings with nonnegative Jacobians, Michigan Math. J., 39 (1992), 495–508.MathSciNetCrossRefGoogle Scholar
  34. [34]
    F. Morgan, Geometric Measure Theory, a Beginner’s Guide, second edition, Academic Press, Boston, 1995.MATHGoogle Scholar
  35. [35]
    M. Ohtsuka, Extremal length and precise functions, Gakkōtosho Co., Tokyo, 2003.MATHGoogle Scholar
  36. [36]
    C. Pommeremke, Boundary Behavior of Conformal Maps, Springer Verlag, Berlin, 1992.CrossRefGoogle Scholar
  37. [37]
    A. Rasila and M. Vuorinen, Experiments with moduli of quadrilaterals, Rev. Roumaine Math. Pures Appl. 51 (2006), 747–757.MathSciNetMATHGoogle Scholar
  38. [38]
    E. Reich, Steiner symmetrization and the conformal moduli of parallelograms, Analysis and Topology, World Sci. Publ., River Edge, NJ, 1998, pp. 615–620.MATHGoogle Scholar
  39. [39]
    E. Reich and H. Walczak, On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc., 117 (1965), 338–351.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    B. Rodin, The method of extremal length, Bull. Amer. Math. Soc., 80 (1974), 587–606.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    B. Rodin and S. E. Warschawski, Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 2 (1976), 467–500.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    B. Rodin and S. E. Warschawski, Extremal length and univalent functions. I. The angular derivative, Math. Z., 153 (1977), 1–17.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    V. A. Shlyk, Capacity of a condenser and the modulus of a family of separating surfaces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 185 (1990), 168–182, 187; translation in J. Soviet Math. 59 (1992), 1240–1248.MATHGoogle Scholar
  44. [44]
    V. A. Shlyk, On the equality between p-capacity and p-modulus, Sibirsk. Mat. Zh., 34 (1993), no. 6, 216–221; translation in Siberian Math. J. 34 (1993), 1196–1200.MathSciNetMATHGoogle Scholar
  45. [45]
    O. Teichmüller, Untersuchungen über konforme und quasikonforme Abhildungen, Deutsche Math., 3 (1938), 621–678.MATHGoogle Scholar
  46. [46]
    A. Vasil’ev, Moduli of Families of Curves for Conformal and Quasiconformal Mappings, Springer-Verlag, Berlin, 2002.CrossRefMATHGoogle Scholar
  47. [47]
    S. K. Vodopyanov, Regularity of mappings inverse to Sobolev mappings, Mat. Sb., 203 (2012), no. 10, 3–32; translation in Sb. Math. 203 (2012), 1383–1410.MathSciNetCrossRefGoogle Scholar
  48. [48]
    J. Väisälä, On quasiconformal mapping in space, Ann. Acad. Sci. Fenn. Ser. AI 298 (1961).Google Scholar
  49. [49]
    L. I. Volkovyskii, Investigation of the type problem for a simply connected Riemann surface, Trudy Mat. Inst. Steklov. 34 (1950).Google Scholar
  50. [50]
    W. P. Ziemer, Extremal length and conformal capacity, Trans. Amer. Math. Soc., 126 (1967), 460–473.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    W. P. Ziemer, Extremal length and p-capacity, Michigan Math. J., 16 (1969), 43–51.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    W. P. Ziemer, Extremal length as a capacity, Michigan Math. J., 17 (1970), 117–128.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2017

Authors and Affiliations

  • Melkana Brakalova
    • 1
  • Irina Markina
    • 2
  • Alexander Vasil’ev
  1. 1.Department of MathematicsFordham UniversityBronxUSA
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

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