Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
upper and lower solutions; weak solution; evolution triple; compact embedding; distributional derivative; operator of type $(S)_{+}$; operator of type $L-(S)_{+}$; $L$-pseudomonotone operator; multivalued problem; extremal solutions; Zorn’s lemma
upper and lower solutions; weak solution; evolution triple; compact embedding; distributional derivative; operator of type $(S)_{+}$; operator of type $L-(S)_{+}$; $L$-pseudomonotone operator; multivalued problem; extremal solutions; Zorn’s lemma
Summary:
We consider nonlinear parabolic boundary value problems. First we assume that the right hand side term is discontinuous and nonmonotone and in order to have an existence theory we pass to a multivalued version by filling in the gaps at the discontinuity points. Assuming the existence of an upper solution $\phi $ and of a lower solution $\psi $ such that $\psi \le \phi $, and using the theory of nonlinear operators of monotone type, we show that there exists a solution $x \in [\psi ,\phi ]$ and that the set of all such solutions is compact in $W_{pq}(T)$. For the problem with a Caratheodory right hand side we show the existence of extremal solutions in $[\psi ,\phi ]$.
References:
[1] Aizicovici S., Papageorgiou N. S.: Infinite dimensional parametric optimal control problems. Math. Nachr. 162 (1993), 17–38. MR 1239573 | Zbl 0807.49001
[2] Boccardo L., Murat F., Puel J.-P.: Existence results for some quasilinear parabolic problems. Nonlin. Anal-TMA 13 (1989), 373–392. MR 0987375
[3] Carl S.: On the existence of extremal weak solutions for a class of quasilinear parabolic problems. Diff. Integ. Eqns 6 (1993), 1493–1505. MR 1235207 | Zbl 0805.35057
[4] Carl S.: Enclosure of solution for quasilinear dynamic hemivariational inequalities. Nonlin. World 3 (1996), 281–298. MR 1411356
[5] Chang K.-C.: Variational methods for nondifferentiable functions and their applications to partial differential equations. J. Math. Anal. 80 (1981), 102–129. MR 0614246
[6] Chipot M., Rodrigues, J-E.: Comparison and stability of solutions to a class of quasilinear parabolic problems. Proc. Royal Soc. Edinburgh 110 A (1988), 275–285. MR 0974743 | Zbl 0669.35052
[7] Costa D. G., Goncalves J. V. A.: Critical point theory for nondifferentiable functionals and applications. J. Math. Anal. 153 ( 1990), 470–485. MR 1080660 | Zbl 0717.49007
[8] Dancer E. N., Sweers G.: On the existence of maximal weak solution for a semilinear elliptic equation. Diff. Integr. Eqns 2 (1989), 533–540. MR 0996759
[9] Deuel J., Hess P.: Nonlinear parabolic boundary value problems with upper and lower solutions. Israel J. Math. 29 (1978), 92–104. MR 0492636 | Zbl 0372.35045
[10] Dunford N., Schwartz J.: Linear Operators I. Wiley, New York (1958).
[11] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, The Netherlands (1997). MR 1485775 | Zbl 0887.47001
[12] Evans L., Gariepy R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). MR 1158660 | Zbl 0804.28001
[13] Halidias N., Papageorgiou N. S.: Second order multivalued boundary value problems. Archivum Math. (Brno) 34 (1998), 267–284. MR 1645320 | Zbl 0915.34021
[14] Kandilakis D., Papageorgiou N. S.: Nonlinear periodic parabolic problems with nonmonotone discontinuities. Proc. Edinburgh Math. Soc. 41 (1998), 117–132. MR 1604345 | Zbl 0909.35074
[15] Kesavan S.: Topics in Functional Analysis and Applications. Wiley, New York (1989). MR 0990018 | Zbl 0666.46001
[16] Lions J.-L.: Quelques Methodes de Resolution des Problems aux Limits Non-Lineaires. Dunod, Paris (1969). MR 0259693
[17] Miettinen M.: Approximation of hemivariational inequalities and optimal control problem. Univ. of Jyvaskyla, Math. Department, Finland, Report 59 (1993). MR 1248824
[18] Miettinen M.: A parabolic hemivariational inequality. Nonl. Anal-TMA 26 (1996), 725–734. MR 1362746 | Zbl 0858.35072
[19] Mokrane A.: Existence of bounded solutions for some nonlinear parabolic equations. Proc. Royal Soc. Edinburgh 107 (1987), 313–326. MR 0924524
[20] Panagiotopoulos P. D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer Verlag, New York, Berlin (1994). MR 1385670
[21] Rauch J.: Discontinuous semilinear differential equations and multiple-valued maps. Proc. AMS 64 (1977), 272–282. MR 0442453 | Zbl 0413.35031
[22] Stuart C.: Maximal and minimal solutions of elliptic equations with discontinuous nonlinearities. Math. Zeitschrift 163 (1978), 239–249. MR 0513729
[23] Zeidler E.: Nonlinear Functional Analysis and its Applications. Springer Verlag, New York (1990). Zbl 0684.47029
Search
Partner of
