On the First Eigenvalue of the Degenerate \(\varvec{p}\)-Laplace Operator in Non-convex Domains

  • V. Gol’dshtein
  • V. Pchelintsev
  • A. Ukhlov


In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, \(p>2\), in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincaré–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.


Elliptic equations Sobolev spaces Composition operators 

Mathematics Subject Classification

35P15 46E35 30C65 



The first author was supported by the United States-Israel Binational Science Foundation (BSF Grant No. 2014055). The second author was supported by RFBR Grant No. 18-31-00011.


  1. 1.
    Ahlfors, L.: Quasiconformal reflections. Acta Math. 109, 291–301 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ahlfors, L.: Lectures on quasiconformal mappings. D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London (1966)Google Scholar
  3. 3.
    Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues, Spectral theory and geometry. (Edinburgh, 1998), 95–139, London Math. Soc. Lecture Note Ser., 273, Cambridge University Press, Cambridge (1999)Google Scholar
  4. 4.
    Ashbaugh, M.S., Benguria, R.D.: Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions. SIAM J. Math. Anal. 24, 557–570 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ashbaugh, M.S., Levine, H.A.: Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres. Journées ”Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1997), Exp. No. I, 15 pp., École Polytech., Palaiseau (1997)Google Scholar
  6. 6.
    Astala, K.: Area distortion of quasiconformal mappings. Acta Math. 173, 37–60 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Astarita, G., Marrucci, G.: Principles of Non-Newtonian Fluids Mechanics. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  8. 8.
    Avkhadiev, F.G.: Introduction to the Geometric Function Theory. Springer, Kazan (2012)Google Scholar
  9. 9.
    Beardon, A.F., Minda, D.: The Hyperbolic Metric and Geometric Function Theory, Quasiconformal Mappings and Their Applications, pp. 9–56. Narosa, New Delhi (2007)zbMATHGoogle Scholar
  10. 10.
    Becker, J., Pommerenke, Ch.: ‘ Hölder continuity of conformal maps with quasiconformal extension. Complex Variab. Theory Appl. 10, 267–272 (1988)zbMATHGoogle Scholar
  11. 11.
    Burenkov, V.I., Gol’dshtein, V., Ukhlov, A.: Conformal spectral stability for the Dirichlet-Laplace operator. Math. Nachr. 288, 1822–1833 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Enache, C., Philippin, G.A.: On some isoperimetric inequalities involving eigenvalues of symmetric free membranes. ZAMM Z. Angew. Math. Mech. 95, 424–430 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20, 253–264 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fait, M., Krzyz, Y., Zygmunt, Y.: Explicit quasiconformal extension for some classes of univalent functions. Comment Math. Helv. 51(2), 279–285 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ferone, V., Nitsch, C., Trombetti, C.: A remark on optimal weighted Poincaré inequalities for convex domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23, 467–475 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York (1977)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gol’dshtein, V.M.: The degree of summability of generalized derivatives of quasiconformal homeomorphisms. Siberian Math. J. 22(6), 821–836 (1981)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gol’dshtein, V., Gurov, L.: Applications of change of variables operators for exact embedding theorems. Integral Equ. Oper. Theory 19, 1–24 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Spectral Estimates of the p-Laplace Neumann operator and Brennan’s Conjecture. Boll. Unione Mat. Ital. 11, 245–264 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Integral estimates of conformal derivatives and spectral properties of the Neumann-Laplacian. J. Math. Anal. Appl. 463, 19–39 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gol’dshtein, V.M., Reshetnyak, Y.G.: Qusiconformal Mappings and Sobolev Spaces. Kluwier Academic Publisher, Dordrecht/Boston/London (1990)CrossRefGoogle Scholar
  22. 22.
    Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361, 3829–3850 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gol’dshtein, V., Ukhlov, A.: Brennan’s conjecture for composition operators on Sobolev spaces. Eurasian Math. J. 3(4), 35–43 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Gol’dshtein, V., Ukhlov, A.: Conformal weights and Sobolev embeddings. J. Math. Sci. (N.Y.) 193, 202–210 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gol’dshtein, V., Ukhlov, A.: Brennan’s Conjecture and universal Sobolev inequalities. Bull. Sci. Math. 138, 253–269 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gol’dshtein, V., Ukhlov, A.: Sobolev homeomorphisms and Brennan’s conjecture. Comput. Methods Funct. Theory 14, 247–256 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gol’dshtein, V., Ukhlov, A.: On the first eigenvalues of free vibrating membranes in conformal regular domains. Arch. Rational Mech. Anal. 221, 893–915 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Gol’dshtein, V., Ukhlov, A.: Spectral estimates of the \(p\)-Laplace Neumann operator in conformal regular domains. Trans. A. Razmadze Math. Inst. 170(1), 137–148 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hajlasz, P., Koskela, P.: Isoperimetric inequalities and imbedding theorems in irregular domains. J. Lond. Math. Soc. 58, 425–450 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs. Oxford University Press, Oxford (1993)zbMATHGoogle Scholar
  31. 31.
    Herron, D.A., Meyer, D.: Quasicircles and bounded turning circles modulo bi-Lipschhitz maps. Rev. Mat. Iberoamericana 28(3), 603–630 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Koskela, P., Onninen, J., Tyson, J.T.: Quasihyperbolic boundary conditions and capacity: Poincaré domains. Math. Ann. 323, 811–830 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Krantz, S.G.: Geometric Function Theory. Explorations in Complex Analysis. Birkhäuser Boston Inc, Boston, MA (2006)zbMATHGoogle Scholar
  34. 34.
    Laugesen, R.S., Morpurgo, C.: Extremals of eigenvalues of Laplacians under conformal mapping. J. Funct. Anal. 155, 64–108 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Maz’ya, V.: Sobolev Spaces: With Applications to Elliptic Partial Differential Equations. Springer, Berlin/Heidelberg (2010)Google Scholar
  36. 36.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292 (1960)CrossRefzbMATHGoogle Scholar
  37. 37.
    Poliquin, G.: Principal frequency of the p-Laplacian and the inradius of Euclidean domains. J. Topol. Anal. 7, 505–511 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rohde, S.: Quasicircles modulo bi-Lipschhitz maps. Rev. Mat. Iberoamericana 17, 643–659 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sevodin, M.A.: Univalence conditions in spiral domains. Tr. Semin. Kraev. Zad. 23, 193–200 (1986). (in Russian)MathSciNetGoogle Scholar
  40. 40.
    Sugawa, T.: Quasiconformal extension of strongly spirallike functions. Comput. Methods Funct. Theory 12(1), 19–30 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3, 343–356 (1954)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Ukhlov, A.: On mappings, which induce embeddings of Sobolev spaces. Siberian Math. J. 34, 185–192 (1993)MathSciNetGoogle Scholar
  43. 43.
    Ukhlov, A., Vodop’yanov, S.K.: Mappings with bounded \((P, Q)\)-distortion on Carnot groups. Bull. Sci. Math. 134, 605–634 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Vodop’yanov, S.K., Ukhlov, A.D.: Superposition operators in Sobolev spaces. Russ. Math. Izvestiya VUZ 46, 11–33 (2002)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Weinberger, H.F.: An isoperimetric inequality for the \(n\)-dimensional free membrane problem. Arch. Rat. Mech. Anal. 5, 633–636 (1956)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael
  2. 2.Division for Mathematics and Computer SciencesTomsk Polytechnic UniversityTomskRussia
  3. 3.Department of General MathematicsTomsk State UniversityTomskRussia

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