Non-Symmetric Perturbation of Symmetric Eigenvalue Problems

In this Chapter we establish the influence of an arbitrary small perturbation for several classes of symmetric hemivariational eigenvalue inequalities with constraints. If the symmetric problem has infinitely many solutions we show that the number of solutions of the perturbed problem tends to infinity if the perturbation approaches zero with respect to an appropriate topology. This is a very natural phenomenon that occurs often in concrete situations. We illustrate it with the following elementary example: consider on the real axis the equation sin x = 1/2. This is a “symmetric” problem (due to the periodicity) with infinitely many solutions. Let us now consider an arbitrary non-symmetric “small” perturbation of the above equation. For instance, the equation sin x = 1/2 + εx2 has finitely many solutions, for any ε ≠ 0. However, the number of solutions of the perturbed equation becomes greater and greater if the perturbation (that is, |ε|) is smaller and smaller. In contrast with this elementary example, our proofs rely on powerful tools such as topological methods in nonsmooth critical point theory. For different perturbation results and their applications we refer to [1], [15], [20] (see also [9] for a nonsmooth setting) in the case of elliptic equations, [8] for variational inequalities and [3], [5], [6], [14], [16], [17], [18] for various perturbations of hemivariational inequalities. This abstract developments are motivated by important appications in Mechanics (see [12], [13]).