Location of Solutions for General Nonsmooth Problems
The aim of the present Chapter is to study from a qualitative point of view a general eigenvalue problem associated to a variational-hemivariational inequality with a constraint for the eigenvalue. The basic feature of our approach is that we are mainly concerned with the location of eigensolution (u, λ), where u and λ stand for the eigenfunction and the eigenvalue, respectively. This is done in Section 2, where the location of eigensolutions is achieved by means of the graph of the derivative of a C 1 function. Section 1 presents a general existence result for variationalhemivariational inequalities with assumptions of Ambrosetti and Rabinowitz type. Section 2 deals with the exposition of our abstract location results. In Section 3 we discuss the location of solutions to variationalhemivariational inequalities by applying the abstract results. The case of nonlinear Dirichlet boundary value problems is contained.
KeywordsEigenvalue Problem Variational Method Generalize Gradient Real Hilbert Space Critical Point Theory
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- H. Brézis and L. Nirenberg, Functional Analysis and Applications to Partial Differential Equations, in preparation.Google Scholar
- D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
- D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume II: Unilateral Problems, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
- J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Nonconvex Optimization and Its Applications, Vol. 35, Dordrecht/Boston/London, 1999.Google Scholar
- J. Haslinger and D. Motreanu, Hemivariational inequalities with a general growth condition: existence and approximation, Appl. Anal., to appear.Google Scholar
- E. S. Mistakidis and G. E. Stavroulakis, Nonconvex Optimization in Mechanics, Nonconvex Optimization and Its Applications, Vol. 21, Kluwer Academic Publishers, Dordrecht/Boston/London, 1998.Google Scholar
- D. Motreanu: Nonsmooth/Nonconvex Mechanics: Modeling, Analysis and Numerical Methods, A Volume dedicated to the memory of Professor P.D. Panagiotopoulos, D. Gao, R. W. Ogden and G. E. Stavroulakis (eds.), Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.Google Scholar
- D. Motreanu, Existence and multiplicity results for variational-hemivariational inequalities in the sense of P. D. Panagiotopoulos, in: Proceedings of the International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, In memoriam of Professor P. D. Panagiotopoulos, 5–6 July 2002, Thessaloniki, Greece, pp. 23–30.Google Scholar
- D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications, Kluwer Academic Publishers, Nonconvex Optimization and Its Applications, Vol. 29, Dordrecht/Boston/London, 1999.Google Scholar
- Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York (1995).Google Scholar
- P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, New York, 1993.Google Scholar
- P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, R.I., 1996.Google Scholar