Optimal regularity and free boundary regularity for the Signorini problem

A proof of the optimal regularity and free boundary regularity is announced and informally discussed for the Signorini problem for the Lamé system. The result, which is the first of its kind for a system of equations, states that if $\mathbf u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb R^3)$ minimizes
$$ J(\mathbf u)=\int_{B_1^+}|\nabla\mathbf u+\nabla^\bot \mathbf u|^2+\lambda(\operatorname{div}(\mathbf u))^2 $$
in the convex set
\begin{align*} K=\big\{\mathbf u&=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb R^3);\; u^3\ge0\textrm{ on }\Pi,\\ \mathbf u&=f\in C^\infty(\partial B_1)\textrm{ on }(\partial B_1)^+\big\}, \end{align*}
where, say, $\lambda\ge0$, then $\mathbf u\in C^{1,1/2}(B_{1/2}^+)$. Moreover, the free boundary, given by $\Gamma_\mathbf u=\partial\{x;\,u^3(x)=0,\,x_3=0\}\cap B_1$, will be a $C^{1,\alpha}$-graph close to points where $\mathbf u$ is nondegenerate. Historically, the problem is of some interest in that it is the first formulation of a variational inequality. A detailed version of this paper will appear in the near future.