Unilateral eigenvalue problems for nonlinearly elastic plates: An approach via pseudo-monotone operators
The paper concerns abstract variational inequalities of the type \[ u\in U,\;\lambda\in\mathbb{R}_+: \langle Au- \lambda L(u)+ C(u)- f, v-u\rangle\geq 0\text{ for all }v\in K.\tag{1} \] Here \(K\) is a closed convex cone in a real Hilbert space \(X\) with scalar product \(\langle\cdot,\cdot\rangle\), \(U\subset K\) is bounded and open, \(A: X\to X\) is linear, bounded and coercive, \(L: K\to X\) (resp. \(C:K\to X\)) is continuous and positively homogeneous of order one (resp. three), and \(\langle C(u),u\rangle> 0\) for all \(u\in K\setminus\{0\}\). Using topological \(S\)-degree, it is proved that, if \(\langle f,u_0\rangle> 0\) for certain \(u_0\in K\), then for all \(\lambda\in \mathbb{R}_+\) there exists a solution to (1). The result is applied to von Karman's equations for a thin elastic plate, clamped on a part of its boundary and simply supported on the remaining part, with one-sided obstacle zero everywhere in the domain.
scientific article; zbMATH DE number 1462257