Opuscula Math. 39, no. 2 (2019), 281-296

Opuscula Mathematica

Extremal length and Dirichlet problem on Klein surfaces

Monica Roşiu

Abstract. The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.

Keywords: Klein surface, extremal length, extremal distance.

Mathematics Subject Classification: 30F50, 35J05, 31A15.

Full text (pdf)

  1. L.V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.
  2. L. Ahlfors, A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101-129.
  3. L.V. Ahlfors, L. Sario, Riemann Surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960.
  4. N.L. Alling, N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math. 219, Springer-Verlag, 1971.
  5. C. Andreian Cazacu, On morphisms of Klein surfaces, Rev. Roumaine Math. Pures Appl. 31 (1986) 6, 461-470.
  6. S.S. Antman, Fundamental Mathematical Problems in the Theory of Nonlinear Elasticity, North-Holland, (1976), 33-54.
  7. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337-403.
  8. I. Bârză, Integration on Nonorientable Riemann Surfaces, [in:] Almost Complex Structures, World Scientific, Singapore-New Jersey-London-Hong Kong, 1995, 63-97.
  9. I. Bârză, D. Ghişa, Explicit formulas for Green's functions on the annulus and on the Möbius strip, Acta Applicandae Mathematicae 54 (1998), 289-302.
  10. E. Bujalance, J.J. Etayo, J.M. Gamboa, G. Gromadzki, Automorphisms Groups of Compact Bordered Klein Surfaces, A Combinatorial Approach, Lecture Notes in Math., vol. 1439, Springer-Verlag, 1990.
  11. J. Jenkins, Univalent functions and conformal mapping, Springer, Berlin, 1958, 13-14.
  12. S.G. Krantz, Partial Differential Equations and Complex Analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
  13. R. Nevanlinna, Analytic Functions, Springer-Verlag, Berlin, 1970.
  14. Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, Cambridge, 2005.
  15. B. Rodin, L. Sario, Principal functions, Van Nostrand, Princeton, N.J., 1968.
  16. M. Roşiu, Harmonic measures and Poisson kernels on Klein surfaces, Electronic Journal of Differential Equations 2017 (2017) 269, 1-7.
  17. L. Sario, K. Oikawa, Capacity functions, Die Grundlehren der math. Wissenschaften, Band 149, Springer-Verlag, New York, 1969.
  18. M. Schiffer, D. Spencer, Functionals of Finite Riemann Surfaces, Princeton University Press, 1954.
  19. M. Seppala, T. Sorvali, Geometry of Riemann Surfaces and Teichmüller Spaces, North-Holland Mathematics Studies, vol. 169, North-Holland, Amsterdam, 1992.
  20. J. Serrin, Removable singularities of solutions of elliptic equations, Arch. Rational Mech. Anal. 17 (1964), 67-78.
  21. J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302.
  22. O. Teichmuller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No. 22, 1-197, 1940. English translation by G. Théret, Extremal quasiconformal mappings and quadratic differentials [in:] Handbook of Teichmuller theory, A. Papadopoulos (ed.), vol. V, EMS Publishing House, 321-484, 2015.
  • Communicated by Marius Ghergu.
  • Received: 2018-10-16.
  • Accepted: 2018-11-03.
  • Published online: 2018-12-07.
Opuscula Mathematica - cover

Cite this article as:
Monica Roşiu, Extremal length and Dirichlet problem on Klein surfaces, Opuscula Math. 39, no. 2 (2019), 281-296, https://doi.org/10.7494/OpMath.2019.39.2.281

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.