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 $J(\u)=\int_{B_1^+}|\nabla \u+\nabla^\bot \u|^2+λ÷(\u)^2$ in the convex set $K=\big\{\u=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\R^3);\; u^3\ge 0 \textrm{on}Π,$ $\u=f\in C^\infty(\partial B_1) \textrm{on}(\partial B_1)^+ \big\},$ where $λ\ge 0$ say. Then $\u\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by $$ Γ_\u=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $$ will be a $C^{1,α}$ 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.

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