Skip to main content
SpringerLink
Log in

Elliptic Problems on the Sierpinski Gasket

  • Chapter
  • First Online:
Topics in Mathematical Analysis and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 94))

Abstract

There are treated nonlinear elliptic problems defined on the Sierpinski gasket, a highly non-smooth fractal set. Even if the structure of this fractal differs considerably from that of (open) domains of Euclidean spaces, this note emphasizes that PDEs defined on it may be studied (as in the Euclidean case) by means of certain variational methods. Using such methods, and appropriate abstract multiplicity theorems, there are proved several results concerning the existence of multiple (weak) solutions of Dirichlet problems defined on the Sierpinski gasket.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arcoya, D., Carmona, J.: A nondifferentiable extension of a theorem of Pucci and Serrin and applications. J. Differ. Equ. 235, 683–700 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonanno, G., Bisci, G.M., Rădulescu, V.: Infinitely many solutions for a class of nonlinear elliptic problems on fractals. C. R. Math. Acad. Sci. Paris 350(3–4), 187–191 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonanno, G., Bisci, G.M., Rădulescu, V.: Variational analysis for a nonlinear elliptic problem on the Sierpinski gasket. ESAIM Control Optim. Calc. Var. 18, 941–953 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Breckner, B.E., Varga, Cs.: One-parameter Dirichlet problems on the Sierpinski gasket. Appl. Math. Comput. 219, 1813–1820 (2012)

    Google Scholar 

  5. Breckner, B.E., Varga, Cs.: Multiple solutions of Dirichlet problems on the Sierpinski gasket. J. Optim. Theory Appl. 1–20 (2013). doi:10.1007/s10957-013-0368-7

    Google Scholar 

  6. Breckner, B.E., Repovš, D., Varga, Cs.: On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket. Nonlinear Anal. 73, 2980–2990 (2010)

    Google Scholar 

  7. Breckner, B.E., Rădulescu, V., Varga, Cs.: Infinitely many solutions for the Dirichlet problem on the Sierpinski Gasket. Anal. Appl. (Singap.) 9, 235–248 (2011)

    Google Scholar 

  8. Falconer, K.J.: Semilinear PDEs on self-similar fractals. Commun. Math. Phys. 206, 235–245 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Hoboken (2003)

    Book  Google Scholar 

  10. Falconer, K.J., Hu, J.: Non-linear elliptical equations on the Sierpinski gasket. J. Math. Anal. Appl. 240, 552–573 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Faraci, F., Kristály, A.: One-dimensional scalar field equations involving an oscillatory nonlinear term. Discrete Continuous Dyn. Syst. 18(1), 107–120 (2007)

    Article  MATH  Google Scholar 

  12. Hu, C.: Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket. Sci. China Ser. A 47(5), 772–786 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hua, C., Zhenya, H.: Semilinear elliptic equations on fractal sets. Acta Math. Sci. Ser. B Engl. Ed. 29 B(2), 232–242 (2009)

    Google Scholar 

  14. Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kigami, J.: A harmonic calculus on the Sierpinski spaces. Jpn. J. Appl. Math. 6, 259–290 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)

    Google Scholar 

  17. Kigami, J.: In quest of fractal analysis. In: Yamaguti, H., Hata, M., Kigami, J. (eds.) Mathematics of Fractals, pp. 53–73. American Mathematical Society, Providence (1993)

    Google Scholar 

  18. Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  19. Kozlov, S.M.: Harmonization and homogenization on fractals. Commun. Math. Phys. 153, 339–357 (1993)

    Article  MATH  Google Scholar 

  20. Kristály, A., Motreanu, D.: Nonsmooth Neumann-type problems involving the p-Laplacian. Numer. Funct. Anal. Optim. 28(11–12), 1309–1326 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977)

    Google Scholar 

  22. Marcus, M., Mizel, V.: Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33, 217–229 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mosco, U.: Variational fractals. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25(3–4), 683–712 (1997)

    Google Scholar 

  24. Mosco, U.: Dirichlet forms and self-similarity. In: Josta, J., et al. (eds.) New Directions in Dirichlet Forms. International Press, Cambridge (1998)

    Google Scholar 

  25. Mosco, U.: Lagrangian metrics on fractals. Proc. Symp. Appl. Math. 54, 301–323 (1998)

    Article  MathSciNet  Google Scholar 

  26. Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000)

    Google Scholar 

  28. Ricceri, B.: A further three critical points theorem. Nonlinear Anal. 71, 4151–4157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ricceri, B.: A further refinement of a three critical points theorem. Nonlinear Anal. 74, 7446–7454 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ricceri, B.: Another four critical points theorem. In: Proceedings of the Seventh International Conference on Nonlinear Analysis and Convex Analysis (NACA’11), Busan, pp. 163–171. Yokohama Publishers, Yokohama (2011)

    Google Scholar 

  31. Ricceri, B.: Addendum to “A further refinement of a three critical points theorem” [Nonlinear Anal. 74, 7446–7454 (2011)]. Nonlinear Anal. 75, 2957–2958 (2012)

    Google Scholar 

  32. Rogers, C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970)

    MATH  Google Scholar 

  33. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw Hill, New York (1987)

    MATH  Google Scholar 

  34. Saint Raymond, J.: On the multiplicity of solutions of the equation \(-\bigtriangleup u =\lambda \cdot f(u)\). J. Differ. Equ. 180, 65–88 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Strichartz, R.S.: Some properties of Laplacians on fractals. J. Funct. Anal. 164, 181–208 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Strichartz, R.S.: Solvability for differential equations on fractals. J. Anal. Math. 96, 247–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    Google Scholar 

  38. Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. III. Springer, New York (1985)

    Book  Google Scholar 

  39. Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. II/A. Springer, New York (1990)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Kogălniceanu str. 1, 400084, Cluj-Napoca, Romania

    Brigitte E. Breckner & Csaba Varga

Authors
  1. Brigitte E. Breckner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Csaba Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brigitte E. Breckner .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics, University of Pécs, Pécs, Hungary

    László Tóth

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Breckner, B.E., Varga, C. (2014). Elliptic Problems on the Sierpinski Gasket. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation