Abstract
The purpose of this chapter is to present several variants of minimization and mountain-pass theorems for nondifferentiable functionals. We assume that the given functionals are locally Lipschitz so that their generalized gradients can be defined (cf. Clarke [Cl] ). A general critical point theory for locally Lipschitz functionals was developed by K. C. Chang [Ch1], extending the concept of a critical point, the Palais—Smale condition and the deformation lemma. Critical point results have also been obtained in other nondifferentiable settings. We refer to Degiovanni & Marzocchi [DM], Corvellec, Degiovanni & Marzocchi [CDM] for the case of continuous functionals and to Ribarska, Tsachev & Krastanov [RTK1], Ioffe & Schwartzman [IS] for the case of discontinuous functionals.
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Bibliography
Adly S, Goeleven D. Periodic solutions for a class of hemivariational inequalities. Comm. Appl. Nonl. Anal., 1995; 2:45–57.
Adly S, Goeleven D. Homoclinic orbits for a class of hemivariational inequalities. Applicable Analysis, 1995; 58:229–240.
Aubin, Jean-Pierre. L ’analyse Non Linéaire et ses Motivations Economique. Paris: Masson, 1984.
Aubin, Jean-Pierre and Ekeland, Ivar. Applied Nonlinear Analysis. N.Y.: John Wiley amp; Sons, 1984.
Brezis H, Nirenberg L. Remarks on finding critical points. Comm. Pure Appl. Math., 1991;XLIV:939–963.
Caklovic L, Li S, Willem M. A note on Palais-Smale condition and coercivity. Diff. Int. Eq., 1990;3:799–800
Chabrowski, Jan. Variational Methods for Potential Equations: with Applications to Nonlinear Elliptic Equations. Berlin: de Gruyter, 1997.
Chang KC. Variational methods for non-differentiable functionals and their applications to PDE. J. Math. Anal. Appl., 1981;80:102–129.
Clarke, Frank. Optimization and Nonsmooth Analysis. N.Y.: John Wiley & Sons, 1983.
Corvellec JN, Degiovanni M, Marzocchi M. Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlin. Anal. 1993;1:151-.
Costa DG, Gonçalves JVA. Critical point theory for nondifferentiable functionals and applications. J. Math. Anal. Appl., 1990;153:470–485.
Degiovanni M., Marzocchi M., A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl., 1994;167:73–100.
Ekeland, Ivar. Convexiy Methods in Hamiltoniam Mechanics. N.Y.: Springer-Verlag, 1990.
Ghoussoub N, Preiss D. A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincaré, 1989;6,5:321–330.
Grossinho MR, Tersian S. Critical point theory for locally Lipschitz functionals and applications to fourth order problems. Proceedings of XXVIII-th Spring Conference of U.B.M., Sofia, 1999:99–106.
Ioffe A., Schwartzman E., Metric critical point theory, 1. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl., 1996;75:125–153.
Ribarska N., Tsachev T. and Krastanov M. The intrinsic mountain pass principle. C. R. Acad. Sci. Paris, Ser I, 1999;329:350–358.
Ribarska N., Tsachev T. and Krastanov M. Deformation Lemma, Ljusternik-Schnirelmann Theory and Mountain Pass theorem on C1-Finsler Manifolds. Serdica Math. J., 1991;21:239–266.
Ribarska N., Tsachev T. and Krastanov M. On the general mountain pass principle of Ghoussoub-Preiss. Mathematica Balkanika (N.S.), 1991;5:399–404.
Panagiotopoulos, Panagiotis. Hemivariational Inequalities. Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993.
Panagiotopoulos PD. Nonconvex energy function, hemivariational inequalities and substationary principles. Acta Mech. 1983;48:160–183.
Panagiotopoulos PD. Coercive and semicoercive hemivariational inequalities. Nonlinear Anal., 1991;16:209–231.
Willem M. Lecture notes on critical point theory, Fundação Universidade de Brasília, 199, 1983.
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do Rosário Grossinho, M., Tersian, S.A. (2001). Minimax Theorems for Locally Lipschitz Functionals and Applications. In: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Nonconvex Optimization and Its Applications, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3308-2_6
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DOI: https://doi.org/10.1007/978-1-4757-3308-2_6
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