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On the existence of two branches of bifurcation for eigenvalue problems associated with variational inequalities. (English) Zbl 0828.58007

Manara, Carlo Felice (ed.) et al., Papers in honor of Giovanni Melzi. Milano: Univ. Cattolica del Sacro Cuore. Sci. Mat. 11, 35-72 (1994).
In the paper the following “variational bifurcation” problem is considered. Let \(H\) be a Hilbert space, \(W\) a convex subset of \(H\), \(f : \overline W \to\;] - \infty, + \infty]\) a function, and \(L : H \to H\) a symmetric linear bounded operator. The bifurcation problem for \(L\) and \(f\) is then \[ \begin{cases} (\lambda,u) \in \mathbb{R} \times W \\ \lambda Lu \in \partial^- f(u) \end{cases} \tag{1} \] where the subdifferential \(\partial^-f\) is appropriately defined. The function \[ f_0 (u) = \Gamma \lim_{\varepsilon \to 0^+} \varepsilon^{-2} f(\varepsilon u) \] is also defined, so that the “linearized problem” is written as \[ \begin{cases} (\lambda,u) \in \mathbb{R} \times H \\ \lambda Lu \in \partial^- f_0 (u). \end{cases} \tag{2} \] It is shown that every bifurcation value of (1) is an eigenvalue of (2). The converse is not true in general; counterexamples can be found in Ann. Mat. Pura Appl., IV. Ser. 156, 37-71 (1990; Zbl 0722.58013) and Ann. Fac. Sci. Toulouse, V. Sér., Math. 11, No. 1, 39-66 (1990; Zbl 0717.49010) by the second author and in Comment. Math. Univ. Carol. 24, No. 4, 657-665 (1983; Zbl 0638.49020) by E. Miersemann. However, the converse holds when \(f_0\) behaves like a quadratic form, and in the paper it is proved that in this case there are at least two branches of bifurcation.
In Section 4 an application to variational inequalities is given.
For the entire collection see [Zbl 0802.00005].
Reviewer: G.Buttazzo (Pisa)

MSC:

58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
49J45 Methods involving semicontinuity and convergence; relaxation
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
49J40 Variational inequalities
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