Skip to main content
SpringerLink
Log in

Opial Inequalities in Several Independent Variables

  • Chapter
Opial Inequalities with Applications in Differential and Difference Equations

Part of the book series: Mathematics and Its Applications ((MAIA,volume 320))

  • 202 Accesses

Abstract

In the year 1981 (April 26 – May 2) during the General Inequalities 3 meeting at Oberwolfach, Agarwal proved a two-independent variable analog of the inequality (1.1.1). This result can be stated as follows: If u(t,s) ∈ C (1,1)([a,T] × [c, S]), u(a, s) = u(t, c) = 0, then

$$ \int_a^T {\int_c^S {\left| {u(t,s){u_{ts}}(t,s)} \right|} } {\text{ }}dt{\text{ }}ds \leqslant {\text{ }}C(T - a)(S - c){\int_a^T {\int_c^S {\left| {{u_{ts}}(t,s)} \right|} } ^2}dt{\text{ }}ds, $$
(4.1.1)

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, R.P., An integro-differential inequality, General Inequalities III, eds. E.F.Beckenbach and W.Walter, Birkhäuser, Basel, 1983, 501–503.

    Google Scholar 

  2. Agarwal, R.P. and P.Y.H. Pang, Sharp Opial-type inequalities in two variables, Applicable Analysis (to appear).

    Google Scholar 

  3. Agarwal, R.P. and P.Y.H. Pang, Opial-type inequalities involving higher order partial derivatives of two functions, (to appear).

    Google Scholar 

  4. Agarwal, R.P. and Qin Sheng, Sharp integral inequalities in n independent variables, Nonlinear Analysis (to appear).

    Google Scholar 

  5. Chen, M.S. and C.T. Lin, A note on Opial’s inequality, Chinese J. Math. 14 (1986), 53–58.

    MathSciNet  MATH  Google Scholar 

  6. Cheung, W.S., On Opial-type inequalities in two variables, Aequationes Math. 38(1989), 236–244.

    Google Scholar 

  7. Cheung, W.S., Some generalized Opial-type inequalities, J. Math. Anal. Appl. 162 (1991), 317–321.

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheung, W.S., Generalizations of Opial-type inequalities in two variables, Tamkang J. Math. 22 (1991), 43–50.

    MathSciNet  MATH  Google Scholar 

  9. Cheung, W.S., Opial-type inequalities with m functions in n variables, Mathematika 39 (1992), 319–326.

    Article  MathSciNet  MATH  Google Scholar 

  10. Crooke, P.S., On two inequalities of the Sobolev type, Applicable Analysis 3 (1974), 345–358.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubinskii, J.A., Some integral inequalities and the solvability of degenerate quasilinear elliptic systems of differential equations, Math. Sb. 64 (1964), 458–480.

    MathSciNet  Google Scholar 

  12. Federer, H. and W. Fleming, Normal and integral currents, Ann. Math. 72 (1960), 458–520.

    Article  MathSciNet  MATH  Google Scholar 

  13. Fleming, W., Functions whose partial derivatives are measures, Illinois J. Math. 4 (1960), 452–478.

    MathSciNet  MATH  Google Scholar 

  14. Fleming, W. and R. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218–222.

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., 1964.

    MATH  Google Scholar 

  16. Friedman, A., Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969.

    Google Scholar 

  17. Garabedian, P.R., Partial Differential Equations, Wiley, New York, 1964.

    MATH  Google Scholar 

  18. Horgan, C.O., Integral bounds for solutions of nonlinear reaction-diffusion equations, J. Appl. Math. Phys. (ZAMP) 28 (1977), 197–204.

    Article  MathSciNet  MATH  Google Scholar 

  19. Horgan, C.O., Plane entry flows and energy estimates for the NavierStokes equations, Arch. Rat. Mech. Anal. 68 (1978), 359–381.

    Article  MathSciNet  MATH  Google Scholar 

  20. Horgan, C.O. and W.E. Olmstead, Exponential decay estimates for a class of nonlinear Dirichlet problems, Arch. Rat. Mech. Anal. 71 (1979), 221–235.

    Article  MathSciNet  MATH  Google Scholar 

  21. Horgan, C.O. and P.R. Nachlinger, On the domain of attraction for steady states in heat conduction, Int. J. Engrg. Sci. 14 (1976), 143–148.

    Article  MathSciNet  MATH  Google Scholar 

  22. Horgan, C.O. and L.T. Wheeler, Spatial decay estimates for NavierStokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978), 97–116.

    Article  MathSciNet  MATH  Google Scholar 

  23. Ladyzhenskaya, O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordan and Beach, New York, 1969.

    MATH  Google Scholar 

  24. Ladyzhenskaya, O.A. and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1969.

    Google Scholar 

  25. Lieberman, G.M., Interior gradient bounds for non-uniformly parabolic equations, Indiana Univ. Math. J. 32 (1983), 519–610.

    Article  MathSciNet  Google Scholar 

  26. Lin, C.T., A note on an integrodifferential inequality, Tamkang J. Math. 23 (1985), 349–354.

    MATH  Google Scholar 

  27. Lin, C.T. and G.S. Yang, A generalized Opial’s inequality in two variables, Tamkang J. Math. 15 (1984), 115–122.

    MathSciNet  MATH  Google Scholar 

  28. Lin, C.T. and G.S. Yang, On some new Opial-type inequalities in two variables, Tamkang J. Math. 17 (1986), 31–36.

    MathSciNet  MATH  Google Scholar 

  29. Mitrinovie, D.S., J.E. Pecarie and A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer, Dordrecht, 1991.

    Book  Google Scholar 

  30. Necaev, I.D., Integral inequalities with gradients and derivatives, Soviet Math. Dokl. 22 (1973), 1184–1187.

    Google Scholar 

  31. Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa 13 (1959), 116–162.

    MathSciNet  Google Scholar 

  32. Pachpatte, B.G., On Opial type inequalities in two independent variables, Proc. Royal Soc. Edinburgh A100 (1985), 263–270.

    Article  MATH  Google Scholar 

  33. Pachpatte, B.G., On Sobolev type integral inequalities, Proc. R. Soc. Edinburgh A103 (1986), 1–14.

    Article  MATH  Google Scholar 

  34. Pachpatte, B.G., On some new integral inequalities in several independent variables, Chinese J. Math. 14 (1986), 69–79.

    MathSciNet  MATH  Google Scholar 

  35. Pachpatte, B.G., A note on new integral inequality, Soochow J. Math. 12 (1986), 63–65.

    MathSciNet  MATH  Google Scholar 

  36. Pachpatte, B.G., On multidimensional integral inequalities involving three functions, Soochow J. Math. 12 (1986), 67–78.

    MathSciNet  MATH  Google Scholar 

  37. Pachpatte, B.G., On Poincaré-type integral inequalities, J. Math. Anal. Appl. 114 (1986), 111–115.

    Article  MathSciNet  MATH  Google Scholar 

  38. Pachpatte, B.G., On two inequalities of the Serrin type, J. Math. Anal. Appl. 116 (1986), 193–199.

    Article  MathSciNet  MATH  Google Scholar 

  39. Pachpatte, B.G., On two inequalities similar to Opial’s inequality in two independent variables, Period. Math. Hung. 18 (1987), 137–141.

    Article  MATH  Google Scholar 

  40. Pachpatte, B.G., A note on Poincaré and Sobolev type integral inequalities, Tamkang J. Math. 18 (1987), 1–7.

    MATH  Google Scholar 

  41. Pachpatte, B.G., On Yang type integral inequalities, Tamkang J. Math. 18 (1987), 89–96.

    MATH  Google Scholar 

  42. Pachpatte, B.G., A note on Sobolev type inequalities in two independent variables, J. Math. Anal. Appl. 122 (1987), 114–121.

    Article  MathSciNet  MATH  Google Scholar 

  43. Pachpatte, B.G., A note on multidimensional integral inequalities, J. Math. Anal. Appl. 122 (1987), 122–128.

    Article  MathSciNet  MATH  Google Scholar 

  44. Pachpatte, B.G., On multidimensional Opial-type inequalities, J. Math. Anal. Appl. 126 (1987), 85–89.

    Article  MathSciNet  MATH  Google Scholar 

  45. Pachpatte, B.G., On some new integral inequalities in two independent variables, J. Math. Anal. Appl. 129 (1988), 375–382.

    Article  MathSciNet  MATH  Google Scholar 

  46. Pachpatte, B.G., On a generalized Opial type inequality in two variables, Anal. sti. Univ. “Al. I. Cuza” din Iasi 35 (1989), 231–235.

    MATH  Google Scholar 

  47. Pachpatte, B.G., On certain two dimensional integral inequalities, Chinese J. Math. 17 (1989), 273–279.

    MathSciNet  MATH  Google Scholar 

  48. Pachpatte, B.G., A note on two dimensional integral inequality, Tamkang J. Math. 21 (1990), 211–213.

    MathSciNet  MATH  Google Scholar 

  49. Pachpatte, B.G., Opial type inequality in several variables, Tamkang J. Math. 22 (1991), 7–11.

    MathSciNet  MATH  Google Scholar 

  50. Pachpatte, B.G., On some variants of Sobolev’s inequality, Soochow J. Math. 17 (1991), 121–129.

    MathSciNet  MATH  Google Scholar 

  51. Payne, L.E., Uniqueness criteria for steady state solutions of the NavierStokes equations, Simp. Int. Appl. Anal. Fis. Mat. (Cagliari-Sassari, 1964 ), 130–153, Edizioni Cremonese, Rome, 1965.

    Google Scholar 

  52. Prenter, P.M., Splines and Variational Methods, Wiley, New York, 1975.

    MATH  Google Scholar 

  53. Rosso, F., Variational methods for pointwise stability of viscous fluid motions, Arch. Rat. ‘Viech. Anal. 86 (1984), 181–195.

    Article  MathSciNet  MATH  Google Scholar 

  54. Serrin, J., The initial value problem for the Navier-Stokes equations, in Nonlinear Problems, ed. R.E. Langer, The Univ. of Wisconsin Press, Madison, 1963, 69–78.

    Google Scholar 

  55. Smith, P.D. and E.W. Stredulinsky, Nonlinear elliptic systems with certain unbounded coefficients, Comm. Pure. Appl. Math. 37 (1984), 495–510.

    Article  MathSciNet  MATH  Google Scholar 

  56. Talenti, G., Best constants in Sobolev’s inequality, Annali di Mat. Pura ed. Appl. 110(1976), 353–372.

    Google Scholar 

  57. Wang, Xu-Jia, Sharp constant in a Sobolev inequality, Nonlinear Analysis, 20 (1993), 261–258.

    Article  MathSciNet  MATH  Google Scholar 

  58. Yang, G.S., Inequality of Opial-type in two variables, Tamkang J. Math. 13 (1982), 255–259.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Kent Ridge, Singapore

    Ravi P. Agarwal & Peter Y. H. Pang

Authors
  1. Ravi P. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Y. H. Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Agarwal, R.P., Pang, P.Y.H. (1995). Opial Inequalities in Several Independent Variables. In: Opial Inequalities with Applications in Differential and Difference Equations. Mathematics and Its Applications, vol 320. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8426-5_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-8426-5_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4524-9

  • Online ISBN: 978-94-015-8426-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation