On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

Michal Křížek; Liping Liu

Applicationes Mathematicae (1996)

  • Volume: 24, Issue: 1, page 97-107
  • ISSN: 1233-7234

Abstract

top
A nonlinear elliptic partial differential equation with the Newton boundary conditions is examined. We prove that for greater data we get a greater weak solution. This is the so-called comparison principle. It is applied to a steady-state heat conduction problem in anisotropic magnetic cores of large transformers.

How to cite

top

Křížek, Michal, and Liu, Liping. "On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type." Applicationes Mathematicae 24.1 (1996): 97-107. <http://eudml.org/doc/219155>.

@article{Křížek1996,
abstract = {A nonlinear elliptic partial differential equation with the Newton boundary conditions is examined. We prove that for greater data we get a greater weak solution. This is the so-called comparison principle. It is applied to a steady-state heat conduction problem in anisotropic magnetic cores of large transformers.},
author = {Křížek, Michal, Liu, Liping},
journal = {Applicationes Mathematicae},
keywords = {comparison principle; anisotropic heat conduction; nonlinear boundary value problem; Newton boundary conditions; anisotropic magnetic cores},
language = {eng},
number = {1},
pages = {97-107},
title = {On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type},
url = {http://eudml.org/doc/219155},
volume = {24},
year = {1996},
}

TY - JOUR
AU - Křížek, Michal
AU - Liu, Liping
TI - On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type
JO - Applicationes Mathematicae
PY - 1996
VL - 24
IS - 1
SP - 97
EP - 107
AB - A nonlinear elliptic partial differential equation with the Newton boundary conditions is examined. We prove that for greater data we get a greater weak solution. This is the so-called comparison principle. It is applied to a steady-state heat conduction problem in anisotropic magnetic cores of large transformers.
LA - eng
KW - comparison principle; anisotropic heat conduction; nonlinear boundary value problem; Newton boundary conditions; anisotropic magnetic cores
UR - http://eudml.org/doc/219155
ER -

References

top
  1. [1] L. Boccardo, T. Gallouët et F. Murat, Unicité de la solution de certaines équations elliptiques non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1159-1164. Zbl0789.35056
  2. [2] L. Čermák and M. Zlámal, Transformation of dependent variables and the finite element solution of nonlinear evolution equations, Internat. J. Numer. Methods Engrg. 15 (1980), 31-40. Zbl0444.65078
  3. [3] P. Doktor, On the density of smooth functions in certain subspaces of Sobolev spaces, Comment. Math. Univ. Carolin. 14 (1973), 609-622. Zbl0268.46036
  4. [4] J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp. 29 (1975), 689-696. Zbl0306.65072
  5. [5] J. Douglas, T. Dupont and J. Serrin, Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form, Arch. Rational Mech. Anal. 42 (1971), 157-168. Zbl0222.35017
  6. [6] M. Feistauer, M. Křížek and V. Sobotíková, An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type, East-West J. Numer. Math. 1 (1993), 267-285. Zbl0835.65128
  7. [7] M. Feistauer and A. Ženíšek, Compactness method in finite element theory of nonlinear elliptic problems, Numer. Math. 52 (1988), 147-163. Zbl0642.65075
  8. [8] J. Franců, Monotone operators. A survey directed to applications to differential equations, Appl. Math. 35 (1990), 257-301. Zbl0724.47025
  9. [9] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. Zbl0289.47029
  10. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977. Zbl0361.35003
  11. [11] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984. Zbl0536.65054
  12. [12] I. Hlaváček and M. Křížek, On a nonpotential and nonmonotone second order elliptic problem with mixed boundary conditions, Stability Appl. Anal. Contin. Media 3 (1993), 85-97. 
  13. [13] I. Hlaváček, M. Křížek and J. Malý, On Galerkin approximations of quasilinear nonpotential elliptic problem of a nonmonotone type, J. Math. Anal. Appl. 184 (1994), 168-189. Zbl0802.65113
  14. [14] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), 1-27. Zbl0708.35019
  15. [15] M. Křížek and Q. Lin, On diagonal dominance of stiffness matrices in 3D, East-West J. Numer. Math. 3 (1995), 59-69. Zbl0824.65112
  16. [16] M. Křížek and P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications, Kluwer, Amsterdam, 1996. Zbl0859.65128
  17. [17] S. Larsson, V. Thomée and N. Y. Zhang, Interpolation of coefficients and transformation of the dependent variable in the finite element methods for the nonlinear heat equation, Math. Methods Appl. Sci. 11 (1989), 105-124. Zbl0663.65118
  18. [18] N. G. Meyers, An example of non-uniqueness in the theory of quasi-linear elliptic equations of second order, Arch. Rational Mech. Anal. 14 (1963), 177-179. Zbl0135.15501
  19. [19] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967. 
  20. [20] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner, Leipzig, 1983. 
  21. [21] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967. Zbl0153.13602
  22. [22] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. 
  23. [23] J. Serrin, On the strong maximum principle for quasilinear second order differential inequalities, J. Funct. Anal. 5 (1970), 184-193. Zbl0188.41701
  24. [24] A. Ženíšek, Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations, Academic Press, London, 1990. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.