Optimal Regularity for The Signorini Problem and its Free Boundary

Abstract

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\breve{=}(u^1,u^2,u^3)\in W^{1,2}(B_1^{+}:{\mathbb{R}}^3)$ minimizes $$J(\breve{)}={\int }_{B_1^{+}}|\nabla \breve{+}{\nabla }^{\perp }{\breve{|}}^2+\lambda ÷({\breve{)}}^2$$ in the convex set $$K=\{\breve{=}(u^1,u^2,u^3)\in W^{1,2}(B_1^{+}:{\mathbb{R}}^3);u^3\ge 0\text{on}\Pi ,$$ $$\breve{=}f\in C^{\infty }(\partial B_1)\text{on}(\partial B_1{)}^{+}\},$$ where $\lambda \ge 0$ say. Then $\breve{\in }{C}^{1,1/2}(B_{1/2}^{+})$. Moreover the free boundary, given by $$ Γ_\u=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $$ will be a $C^{1,\alpha }$ 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.

References in corpus (1)

• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian