Arbitrary-friction incremental contact in three-dimensional isotropic elasticity

Let $\Omega \subset \mathbb{R}^3$ be a bounded connected domain of class $C^{1,1}$ with outward normal $n$, and let $\Gamma_U, \Gamma_T, \Gamma_C$ be disjoint relatively open subsets of $\partial\Omega$ whose closures cover $\partial\Omega$, such that $\Gamma_U$ and $\Gamma_C$ have positive surface measure, $\Gamma_C$ is a boundary patch with Lipschitz boundary, and $\operatorname{dist}(\Gamma_U, \Gamma_C) > 0$. For boundary traces write $v_n = v \cdot n$ and $v_t = v - v_n n$. Let $V = \{v \in H^1(\Omega; \mathbb{R}^3) : v = 0 \text{ on } \Gamma_U\}$ and $\varepsilon(v) = \tfrac12 (\nabla v + \nabla v^T)$. Let $G, \lambda \in W^{1,\infty}(\Omega)$ be isotropic Lamé moduli with $G \ge c$ and $3\lambda + 2G \ge c$ for some $c > 0$, and let $$a(u, v) = \int_\Omega \bigl(2G\, \varepsilon(u) : \varepsilon(v) + \lambda \operatorname{div} u \operatorname{div} v\bigr)\, dx.$$ Given $F \in L^2(\Omega; \mathbb{R}^3)$, $T \in L^2(\Gamma_T; \mathbb{R}^3)$, a gap $g \in H^{1/2}(\Gamma_C)$, a previous tangential displacement $w_t \in H^{1/2}(\Gamma_C; \mathbb{R}^3)$ with $w_t \cdot n = 0$, and a friction coefficient $f \in W^{1,\infty}(\Gamma_C)$ with $f \ge 0$, let $\ell(v) = \int_\Omega F \cdot v \, dx + \int_{\Gamma_T} T \cdot v \, dS$ and let $K$ be the set of $t \in H^{-1/2}(\Gamma_C)$, the dual of $H^{1/2}(\Gamma_C)$, that are nonpositive as distributions. For $t \in K$ let $u_t \in V$ be the minimizer of $$E_t(v) = \tfrac12 a(v, v) - \ell(v) - \langle t, v_n \rangle + \langle -ft, |v_t - w_t| \rangle,$$ the brackets denoting duality pairings on $\Gamma_C$; it exists and is unique. A $t \in K$ with $$\bigl\langle (u_t)_n - g, \hat t - t \bigr\rangle \ge 0 \qquad \text{for every } \hat t \in K$$ gives a solution of one time step of the quasistatic Signorini–Coulomb problem of an elastic body in frictional contact with a rigid obstacle, with $t$ the normal contact reaction.

Eck and Jarušek proved existence of solutions of this time-incremental problem, in dimensions two and three and for slightly more regular data, for friction coefficients below a constant that depends only on the elasticity tensor. Ballard and Iurlano proved existence in dimension two for isotropic elasticity, homogeneous or not, without any smallness condition on $f$; for anisotropic elasticity they proved existence below a critical friction coefficient and gave an example of a steady sliding problem with no solution above it. The one step of their proof that requires dimension two is a compactness property that fails in higher dimension.

Problem. For every such $\Omega$, $\Gamma_U$, $\Gamma_T$, $\Gamma_C$, $G$, $\lambda$, $F$, $T$, $g$, $w_t$ and every friction coefficient $f \ge 0$ as above, with no smallness condition on $f$, is there a $t \in K$ with $\langle (u_t)_n - g, \hat t - t \rangle \ge 0$ for every $\hat t \in K$?