Skip to main content
Log in

Prospects for a central theory of partial differential equations

  • Published:
The Mathematical Intelligencer Aims and scope Submit manuscript

Abstract

three episodes in the history of equation-solving are finding zeros of polynomials, solution of ordinary differential equations, and solutions of partial differential equations. The first two episodes went through a number of phases before reaching a rather satisfactory state. That the third episode might develop similarly is the topic of this note.

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. R. A. Adams,Sobolev Spaces, Academic Press, 1978.

  2. R. Chill and M. A. Jendoubi,Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Analysis 53 (2003), 1017–1039.

    Article  MATH  MathSciNet  Google Scholar 

  3. L. C. Evans,Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser., Amer. Math. Soc. 1990.

    MATH  Google Scholar 

  4. Faragó, I., Karátson, J.,Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications. Advances in Computation, Volume 11, NOVA Science Publishers, New York, 2002.

    Google Scholar 

  5. S.-Z. Huang and P. TakaČ,Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal. 46 (2001), 675–698.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Jaffe and C. Taubes,Vortices and Monopoles. Structure of Static Gauge Theories, Progress in Physics 2, 1980.

  7. S. Lojasiewicz,Une propriété topologique des sous-ensembles analytiques rés, Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Editions du C.N.R.S., Paris (1963), 87–89.

  8. D. Lupo and K. R. Payne,On the maximum principle for generalized solutions ta the Tricomi problem, Commun. Contemp. Math. 2 (2000), 535–557.

    MATH  MathSciNet  Google Scholar 

  9. J. Moser,A Rapidly Convergent Iteration Method and Non-Linear Differential Equations, Ann. Scuola Normal Sup. Pisa, 20 (1966), 265–315.

    MATH  Google Scholar 

  10. John M. Neuberger, A numerical method for finding sign changing solutions of superlinear Dirichlet problems, Nonlinear World 4 (1997), 73–83.

    MATH  MathSciNet  Google Scholar 

  11. J. W. Neuberger,Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics 1670, 1997.

  12. J. W. Neuberger,Continuous Newton’s Method for Polynomials, Mathematical Intelligencer, 21 (1999), 18–23.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. W. Neuberger,A near minimal hypothesis Nash-Moser Theorem, Int. J. Pure. Appl. Math. 4 (2003), 269–280.

    MATH  MathSciNet  Google Scholar 

  14. J. W. Neuberger and R. J. Renka,Numerical determination of vortices in superconductors: simulation of cooling, Supercond. Sci. Technol. 16 (2003), 1–4.

    Article  Google Scholar 

  15. B. Sz-Nagy and F. Riesz,Functional Analysis, Ungar, 1955.

  16. L. Simon,Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. Math. 118 (1983), 525–571.

    Article  MATH  Google Scholar 

  17. S. Sial, J. Neuberger, T. Lookman, A. Saxena,Energy minimization using Sobolev gradients: applications to phase separation and ordering. J. Comp. Physics 189 (2003), 88–97.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Struwe,Variational Methods, Ergebnisse Mathematik u. Grenzgebiete, Springer, 1996.

    Google Scholar 

  19. J. von Neumann,Functional Operators II, Annals. Math. Stud. 22, (1940).

  20. E. Zeidler,Nonlinear Functional Analysis and its Applications III, Springer-Verlag, 1985.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to John W. Neuberger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Neuberger, J.W. Prospects for a central theory of partial differential equations. The Mathematical Intelligencer 27, 47–55 (2005). https://doi.org/10.1007/BF02985839

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02985839

Keywords

Navigation