Double eigenvalue problems for hemivariational inequalities
The paper is devoted to study the double eigenvalue problem consisting in finding \(u_1\), \(u_2\in V\) and \(\lambda_1\), \(\lambda_2\in\mathbb{R}\) such that \[ a_1(u_1,v_1)+ C_1(u_1,v_1)+ a_2(u_1,v_1)+ C_2(u_1,v_1)+ \int_\Omega j^0_y(x,(u_1-u_2)(x); (v_1-v_2)(x))d\Omega \] \[ \geq \lambda_1 (B_1u_1,v_1)_V+ \lambda_2 (B_2u_2,v_2)_V, \quad\forall v_1,v_2\in V, \] where \(j_y^0 (\cdot;\cdot)\) is the directional Clarke derivative of a locally Lipschitz \(j: \Omega \times \mathbb{R}^N \to\mathbb{R}\). This new problem turns out to have many important applications in such nonsmooth mechanical problems like the behaviour of sandwich beams, adhesively connected plates, liquid crystals. The authors have applied appropriate extensions of the nonsmooth critical point theory of Chang and of some minimax methods of Rabinowitz to obtain the general existence result. There is an interesting approach presented based on Ekeland's variational principle to the eigenvalue problem in which \(u_1\), \(u_2\in V\) and \(\lambda_1\), \(\lambda_2\in\mathbb{R}\) are sought such that \[ a_1(u_1,v_1)+ C_1(u_1,v_1)+ a_2(u_1, v_1)+ C_2(u_1,v_1)+ \int_\Omega j^0_y (x,(u_1-u_2) (x);(v_1-v_2) (x))d \Omega \] \[ \geq \lambda_1 (B_1u_1,v_1)_V+ \lambda_2 (B_2u_2,v_2)_V, \quad\forall v_1,v_2\in V, \] provided \[ a(B_1u_1,u_1)_V+ b(B_2u_2,u_2)_V =r^2. \] It gives the possibility to obtain some multiplicity results to the aforementioned eigenvalue problems by means of techniques of Ljusternik-Schnirelman suitably adapted to the setting of hemivariational inequalities. The paper is warmly recommended to researchers dealing with nonsmooth nonconvex variational problems and their applications.
scientific article; zbMATH DE number 1136746