English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $$\mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n)$$u=(u1,u2,⋯,un)∈W1,2(B1+:Rn) minimizes $$\begin{aligned} J(\mathbf{u })=\int _{B_1^+}|\nabla \mathbf{u }+\nabla ^\bot \mathbf{u }|^2+\lambda \hbox {div}(\mathbf{u })^2 \end{aligned}$$J(u)=∫B1+|∇u+∇⊥u|2+λdiv(u)2in the convex set $$\begin{aligned} K= & {} \left\{ \mathbf{u }=(u^1,u^2,\dots ,u^n)\in W^{1,2}(B_1^+:{\mathbb {R}}^n);\; u^n\ge 0 \quad \hbox {on}\quad \Pi ,\right. \\&\left. \mathbf{u }=f\in C^\infty (\partial B_1) \quad \hbox {on}\quad (\partial B_1)^+ \right\} , \end{aligned}$$K=u=(u1,u2,⋯,un)∈W1,2(B1+:Rn);un≥0onΠ,u=f∈C∞(∂B1)on(∂B1)+,where $$\lambda \ge 0$$λ≥0 say. Then $$\mathbf{u }\in C^{1,1/2}(B_{1/2}^+)$$u∈C1,1/2(B1/2+). Moreover the free boundary, given by $$\begin{aligned} \Gamma _\mathbf{u }=\partial \{x;\;u^n(x)=0,\; x_n=0\}\cap B_{1}, \end{aligned}$$Γu=∂{x;un(x)=0,xn=0}∩B1,will be a $$C^{1,\alpha }$$C1,α graph close to points where $$\mathbf{u }$$u 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.

Optimal regularity for the Signorini problem and its free boundary