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Optimal Regularity and free boundary regularity for the Signorini problem
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Andersson, John (2013) Optimal Regularity and free boundary regularity for the Signorini problem. St. Petersburg Mathematical Journal, Volume 24 (Number 3). pp. 371386. doi:10.1090/S106100222013012441
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Official URL: http://dx.doi.org/10.1090/S106100222013012441
Abstract
A proof of the optimal regularity and free boundary regularity is announced and informally discussed for the Signorini problem for the Lamé system. The result, which is the first of its kind for a system of equations, states that if $ \textbf {u}=(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb{R}^3)$ minimizes
$\displaystyle J(\textbf {u})=\int _{B_1^+}\vert\nabla \textbf {u}+\nabla ^\bot \textbf {u}\vert^2+\lambda \big (\operatorname {div}(\textbf {u})\big )^2 $
in the convex set
$\displaystyle K=\big \{ \textbf {u} =(u^1,u^2,u^3)\in W^{1,2}(B_1^+:\mathbb{R}^3);\; u^3\ge 0 \ $$\displaystyle \text { on } \ \Pi ,$
$\displaystyle \textbf {u} =f\in C^\infty (\partial B_1) \ $$\displaystyle \text { on }\ (\partial B_1)^+ \big \},$
where, say, $ \lambda \ge 0$, then $ \textbf {u}\in C^{1,1/2}(B_{1/2}^+)$. Moreover, the free boundary, given by $ \Gamma _\textbf {u}=\partial \{x;\;u^3(x)=0,\; x_3=0\}\cap B_{1}, $ will be a $ C^{1,\alpha }$graph close to points where $ \textbf {u}$ is nondegenerate. Historically, the problem is of some interest in that it is the first formulation of a variational inequality. A detailed version of this paper will appear in the near future.
Item Type:  Journal Article  

Divisions:  Faculty of Science > Mathematics  
Journal or Publication Title:  St. Petersburg Mathematical Journal  
Publisher:  American Mathematical Society  
ISSN:  18712509  
Official Date:  21 March 2013  
Dates: 


Volume:  Volume 24  
Number:  Number 3  
Page Range:  pp. 371386  
DOI:  10.1090/S106100222013012441  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access 
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