## 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 $$\Gamma_2$$).

### MSC:

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

### Keywords:

elastic contact problem