Minimax Theorems for Locally Lipschitz Functionals and Applications
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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.
KeywordsGeneralize Gradient Directional Derivative Convergent Subsequence Critical Point Theory Minimax Theorem
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