Optimal Regularity for The Signorini Problem and its Free Boundary

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3)$ minimizes $$J(\u)=\int_{B_1^+}|\nabla \u+\nabla^\bot \u|^2+\lambda\div(\u)^2$$ in the convex set $$K=\big\{\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3);\; u^3\ge 0 \textrm{on}\Pi,$$ $$\u=f\in C^\infty(\partial B_1) \textrm{on}(\partial B_1)^+ \big\},$$ where $\lambda\ge 0$ say. Then $\u\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by \Gamma_\u=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, will be a $C^{1,\alpha}$ graph close to points where \u is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance \cite{AC} and \cite{ACS}). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.