On higher eigenvalues of variational inequalities. (English) Zbl 0638.49020

Miersemann, E.

Commentat. Math. Univ. Carol. 24, 657-665 (1983).

Various previously published results and theorems on eigenvalue problems for variational inequalities are collected in this short paper. More precisely, let a(u,v) and b(u,v) be bounded, real and symmetric bilinear forms defined on the real Hilbert space H and let K be a closed and convex cone of H. Suppose a(u,v) coercive and b(u,v) completely continuous. The author studies the problem \[ (*)\quad u\in K,\quad a(u,v-u)\geq \mu b(u,v-u)\text{ for all } v\in K, \] where \(\mu\) is a real parameter. The existence of eigenvalues for the problem (*) is proved using a variational approach. This proof forms the original part of the work.
An interesting, although not new, application of the main theorem to the problem of buckling of a clamped circular plate with a unilateral constraint is given in the final part of the paper.

MSC:

49R50	Variational methods for eigenvalues of operators (MSC2000)
49J40	Variational inequalities
74S30	Other numerical methods in solid mechanics (MSC2010)

Keywords:

eigenvalue problems for variational inequalities