Skip to main content
Log in

Clifford Valued Weighted Variable Exponent Spaces with an Application to Obstacle Problems

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

In this paper, we introduce the weighted variable exponent spaces in the context of Clifford algebras. After discussing the properties of these spaces, we obtain the existence of weak solutions for obstacle problems for nondegenerate A-Dirac equations with variable growth in the setting of these spaces. Furthermore, we also obtain the existence and uniqueness of weak solutions to the scalar parts of nondegenerate A-Dirac equations in Dirac Sobolev spaces.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Blaya R.A., Reya J.B.: Duality for Harmonic Differential Forms Via Clifford Analysis. Advances in Applied Clifford Algebras 17(4), 589–610 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brackx F., Delanghe R., Sommen F.: Diffrential Forms and/or Multi-vector Functions in Hermitean Clifford Analysis. Cubo 13, 85–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable exponents. Springer, 2011.

  4. Delanghe R., Sommen F., Soucek V.: Clifford Algebra and Spinor-Valued Function. Kluwer Academic Publishers, Dordrecht (1992)

    Book  MATH  Google Scholar 

  5. C. Doran and A. Lasenby, Geometric Algebra for Physicists. Cambridge University Press, 2003.

  6. Dubinskii J., Reissig M.: Variational Problems in Clifford Analysis. Mathematical Methods in the Applied Sciences 25, 1161–1176 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. M. Eleuteri and J. Habermann, Regularity Results for a Class of Obstacle Problems under Nonstandard Growth Conditions. Journal of Mathematical Analysis and Applications 344 (2008), 1120– 1142.

    Google Scholar 

  8. Edmunds D., Rákosník J.: Sobolev Embedding with Variable Exponent. Studia Math. 143, 267–293 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Edmunds D., Rákosník J.: Sobolev Embedding with Variable Exponent. II. Math. Nachr. 246, 53–67 (2002)

    Article  Google Scholar 

  10. Fu Y.: Weak Solution for Obstacle Problem with Variable Growth. Nonlinear Analysis 59, 371–383 (2004)

    MathSciNet  MATH  Google Scholar 

  11. X. L. Fan and D. Zhao, On The Spaces L p(x) and W m,p(x). Journal of Mathematical Analysis and Applications 263 (2001), 424–446.

  12. X. L. Fan, J. S. Shen and D. Zhao, Sobolev Embedding Theorems for Spaces W k, p(x)(Ω). Journal of Mathematical Analysis and Applications 262 (2001), 749–760.

    Google Scholar 

  13. J. Gilbert and M. A. M. Murray, Clifford Algebra and Dirac Oprators in Harmonic Analysis. Oxford University Press, 1993.

  14. Hudzik H.: The problems of separability, duality, reflexivity and of comparison for generalized orlicz-sobolev spaces \({W^k_m(\Omega)}\) . Comment. Math. Prace Mat 21, 315ΓÇô324 (1979)

    MathSciNet  MATH  Google Scholar 

  15. Harjulehto P., Hästö P., Lê Ú.V., Nuortio M.: Overview of Differential Equations with Non-standard Growth. Nonlinear Analysis 72, 4551–4574 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari and N. Marola, An Obstacle Problem and Superharmonic Functions with Nonstandard Growth. Nonlinear Analysis 67 (2007), 3424–3440.

  17. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Porential Theory of Degenerate Elliptic Equations. Oxford University Press, 1993.

  18. Kováčik O., Rákosník J.: On spaces L p(x) and W k,p(x). Czechoslovak Mathematical Journal 41, 592–618 (1991)

    MathSciNet  Google Scholar 

  19. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, 1980.

  20. C. A. Nolder, A-Harmonic Equations and the Dirac Operator. Journal of Inequality and Applications. 2010, Article ID 124018.

  21. Nolder C.A.: Nonlinear A-Dirac Equations. Advances in Applied Clifford Algebras 21((2), 429–440 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nolder C.A., Ryan J.: p-Dirac Operators. Advances in Applied Clifford Algebras 19(2), 391–402 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nolder C.A.: Conjungate Harmonic Functions and Clifford Algebras. Journal of Mathematical Analysis and Applications 302, 137–142 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, 2000.

  25. J. Ryan and W. Sproessig, Clifford Algebras and Their Applications in Mathematical Physics: Volume 2: Clifford Analysis. Birkhäuser, 2000.

  26. Rodrigues J.F., Sanchon M., Urbano J.M.: The Obstacle Problem for Nonlinear Elliptic Equations with Variable Growth and L 1-data. Monatsh. Math 154, 303–322 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Binlin Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fu, Y., Zhang, B. Clifford Valued Weighted Variable Exponent Spaces with an Application to Obstacle Problems. Adv. Appl. Clifford Algebras 23, 363–376 (2013). https://doi.org/10.1007/s00006-013-0383-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-013-0383-7

Keywords

Navigation