Continuous frictionless elasticity contact is well posed

Given: The real Hilbert space $V$, the nonempty closed convex set $K$, the bounded load(s), and the elasticity form $a$ in the setting above.

Countable Choice is assumed by Stampacchia existence and by the published stability theorem. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

The Axiom of Choice is inherited through the bounded C1 trace and Poincare-Wirtinger suppliers used by the local Korn result. (The Axiom of Choice, Korn coercivity with a partial Dirichlet segment, The partial-trace displacement space is a Hilbert space, Poincare-Wirtinger on bounded connected extension domains by Rellich compactness)

$V$ is Hilbert, and $K\subseteq V$ is nonempty, closed and convex. (The partial-trace displacement space is a Hilbert space, The Signorini contact set is nonempty, closed, and convex)

The form is bounded with $\mathrm{\mid }a(u,v)\mathrm{\mid }\le M\mathrm{\parallel }u{\mathrm{\parallel }}_V\mathrm{\parallel }v{\mathrm{\parallel }}_V$, where $M=2\mu +2\lambda$. (The isotropic contact elasticity form is well defined and bounded)

The form is coercive with $a(v,v)\ge \alpha \mathrm{\parallel }v{\mathrm{\parallel }}_V^2$ and $\alpha =2\mu \mathrm{/}{c}_K^2>0$, because $\mathrm{\parallel }v{\mathrm{\parallel }}_V\le c_K\mathrm{\parallel }\epsilon (v){\mathrm{\parallel }}_2$ and $a(v,v)\ge 2\mu \mathrm{\parallel }\epsilon (v){\mathrm{\parallel }}_2^2$. (Korn coercivity with a partial Dirichlet segment, Small-strain elasticity with a frictionless contact gap)

On a real Hilbert space, a nonempty closed convex set and a bounded coercive bilinear form yield a unique variational-inequality solution for every bounded linear load. (Stampacchia's variational inequality)

For two loads on the same Hilbert space, feasible set, and bounded coercive form, the solutions satisfy $\alpha \mathrm{\parallel }{u}_1-u_2\mathrm{\parallel }\le \mathrm{\parallel }{F}_1-F_2{\mathrm{\parallel }}_{V^{\ast }}$. (Lipschitz stability of strongly monotone variational inequalities)