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Sub-supersolutions in a variational inequality related to a sandpile problem

Published online by Cambridge University Press:  17 February 2009

Vy Khoi Le
Vy Khoi Le
Affiliation:
Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65409, USA; e-mail: [email protected].
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Abstract

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In this paper we study a variational inequality in which the principal operator is a generalised Laplacian with fast growth at infinity and slow growth at 0. By defining appropriate sub-and super-solutions, we show the existence of solutions and extremal solutions of this inequality above the subsolutions or between the sub- and super-solutions.

Keywords

sub-supersolutionsandpile problemvariational inequalityextremal solution.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 2 , October 2006 , pp. 179 - 197
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Adams, R., Sobolev Spaces (Academic Press, New York, 1975).Google Scholar
[2]Aronsson, G., Evans, L. C. and Wu, Y., “Fast/slow diffusion and growing sandpiles”, J. Differential Equations 131 (1996) 304–335.CrossRefGoogle Scholar
[3]Carl, S. and Le, V. K., “Enclosure results for quasilinear systems of variational inequalities”, J. Differential Equations 199 (2004) 77–95.CrossRefGoogle Scholar
[4]Carl, S. and Le, V. K., “Sub-supersolution method for quasilinear parabolic variational inequalities”, J. Math. Anal. Appl. 293 (2004) 269–284.CrossRefGoogle Scholar
[5]Carl, S., Le, V. K. and Motreanu, D., “Existence and comparison results for quasilinear evolution hemivariational inequalities”, Electron. J. Differential Equations 57 (2004) 1–17.Google Scholar
[6]Carl, S., Le, V. K. and Motreanu, D., “The sub-supersolution method and extremal solutions for quasilinear hemivariational inequalities”, Differential Integral Equations 17 (2004) 165–178.CrossRefGoogle Scholar
[7]Carl, S., Le, V. K. and Motreanu, D., “Existence and comparison principles for general quasilinear variational-hemivariational inequalities”, J. Math. Anal. Appl. 302 (2005) 65–83.CrossRefGoogle Scholar
[8]Chipot, M., Variational Inequalities and Flow in Porous Media (Springer, New York, 1984).CrossRefGoogle Scholar
[9]Friedman, A., Variational Principles and Free Boundary Value Problems (Wiley-Interscience, New York, 1983).Google Scholar
[10]Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1983).Google Scholar
[11]Gossez, J. P., “Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients”, Trans. Amer. Math. Soc. 190 (1974) 163–205.CrossRefGoogle Scholar
[12]Kinderlehrer, D. and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980).Google Scholar
[13]Krasnosels'kii, M. A. and Rutic'kii, J., Convex Functions and Orlicz Spaces (Noorhoff, Groningen, 1961).Google Scholar
[14]Kufner, A., John, O. and Fučic, S., Function Spaces (Noordhoff, Leyden, 1977).Google Scholar
[15]Feldman, M., Evans, L. C. and Gariepy, R. F., “Fast/slow diffusion and collapsing sandpiles”, J. Differential Equations 137 (1997) 166–209.Google Scholar
[16]Le, V. K., “Existence of positive solutions of variational inequalities by a subsolution-supersolution approach”, J. Math. Anal. Appl. 252 (2000) 65–90.CrossRefGoogle Scholar
[17]Le, V. K., “Subsolution-supersolution method in variational inequalities”, Nonlinear Analysis 45 (2001) 775–800.Google Scholar
[18]Le, V. K. and Schmitt, K., “Sub-supersolution theorems for quasilinear elliptic problems: A variational approach”, Electron. J. Differential Equations 118 (2004) 1–7.Google Scholar
[19]Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (Dunod, Paris, 1969).Google Scholar
[20]Prigozhin, L., “On the bean critical-state model in superconductivity”, Euro. J. Appl. Math. 7 (1996) 237–247.CrossRefGoogle Scholar
[21]Prigozhin, L., “Variational model of sandpile growth”, Euro. J. Appl. Math. 7 (1996) 225–235.CrossRefGoogle Scholar
[22]Rodrigues, J. F., Obstacle Problems in Mathematical Physics (North-Holland, Amsterdam, 1987).Google Scholar