Signorini problem

Problem

Let $\Omega \subseteq \R^3$ be a bounded open set with Lipschitz boundary, the reference shape of a linearly elastic body. Split the boundary into disjoint parts $\Gamma_D$, $\Gamma_N$ and $\Gamma_C$, with $\Gamma_D$ of positive surface measure, and let $n$ be the outward unit normal. For a displacement $u \colon \Omega \to \R^3$ let $\varepsilon(u) = \tfrac12 (\nabla u + \nabla u^T)$ be the strain and $\sigma(u) = \mathcal{C} \varepsilon(u)$ the stress. Here $\mathcal{C}$ is a bounded elasticity tensor with the usual symmetries that is uniformly positive definite on symmetric matrices. Let $f$ be a body force and $g$ a surface force on $\Gamma_N$. Write $u_n = u \cdot n$, $\sigma_n = (\sigma(u) n) \cdot n$ and $\sigma_T = \sigma(u) n - \sigma_n n$. Find $u$ such that:

1. $-\operatorname{div} \sigma(u) = f$ in $\Omega$;
2. $u = 0$ on $\Gamma_D$;
3. $\sigma(u) n = g$ on $\Gamma_N$;
4. on $\Gamma_C$: $u_n \le 0$, $\sigma_n \le 0$, $\sigma_n u_n = 0$ and $\sigma_T = 0$.

Clause 4 describes frictionless contact with a rigid support. The body cannot enter the support, the support can only push, and it pushes only where there is contact.