Quasivariational inequalities and applications in frictional contact problems with normal compliance. (English) Zbl 0977.47057

This article deals with the problem of finding \(u\in V\) such that \[ (Au, v-u)_V+ j(u, v)- j(u,u)\geq (f,v- u)_V\qquad (v\in V), \] where \(A: V\to V\) satisfies the conditions
(a) there exists \(m>0\) such that \((Au- Av,u-v)_V\geq m\|u-v\|^2_V\) \((u,v\in V)\),
(b) there exists \(M>0\) such that \(\|Au- Av\|_V\leq M\|u-v\|_V\) \((u,v\in V)\),
\(j: V\times V\to \mathbb{R}\) and such that \(j(\eta,\cdot): V\to \mathbb{R}\) is a convex functional on \(V\) for each \(\eta\in V\), \(f\in V\), \(V\) a real Hilbert space.
The main result presents a series conditions under which the problem under consideration has at least a solution, has a unique solution, and has a unique solution which depends Lipschitz continuously on \(f\in V\). As application the elastic contact problem is studied; this problem is described with the following system \[ \sigma= F(\varepsilon(u)),\quad \text{div }\sigma+ \varphi_\ell= 0 \] with some special boundary value conditions. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N= 2,3\) fulfilled with an elastic body; the boundary \(\Gamma\) of \(\Omega\) is regular and partioned into three disjoint measurable parts (the body is clamped on \(\Gamma_1\), a volume force of density \(\varphi_1\) acts in \(\Omega\) and a surface traction of density \(\varphi_2\) acts on \(\Gamma_2\)).


47J20 Variational and other types of inequalities involving nonlinear operators (general)
74M15 Contact in solid mechanics