We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if u=(u1,u2,⋯,un)∈W1,2(B1+:Rn)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n)$$\end{document} minimizes J(u)=∫B1+|∇u+∇⊥u|2+λdiv(u)2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} J(\mathbf{u })=\int _{B_1^+}|\nabla \mathbf{u }+\nabla ^\bot \mathbf{u }|^2+\lambda \hbox {div}(\mathbf{u })^2 \end{aligned}$$\end{document}in the convex set K=u=(u1,u2,⋯,un)∈W1,2(B1+:Rn);un≥0onΠ,u=f∈C∞(∂B1)on(∂B1)+,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} K= & {} \left\{ \mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n);\; u^n\ge 0 \quad \hbox {on}\quad \Pi ,\right. \\&\left. \mathbf{u }=f\in C^\infty (\partial B_1) \quad \hbox {on}\quad (\partial B_1)^+ \right\} , \end{aligned}$$\end{document}where λ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \ge 0$$\end{document} say. Then u∈C1,1/2(B1/2+)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u }\in C^{1,1/2}(B_{1/2}^+)$$\end{document}. Moreover the free boundary, given by Γu=∂{x;un(x)=0,xn=0}∩B1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Gamma _\mathbf{u }=\partial \{x;\;u^n(x)=0,\; x_n=0\}\cap B_{1}, \end{aligned}$$\end{document}will be a C1,α\documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\alpha }$$\end{document} graph close to points where u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{u }$$\end{document} is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.
