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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. 371-386. doi:10.1090/S1061-0022-2013-01244-1 ISSN 1871-2509.
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Official URL: http://dx.doi.org/10.1090/S1061-0022-2013-01244-1
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 | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | St. Petersburg Mathematical Journal | ||||
Publisher: | American Mathematical Society | ||||
ISSN: | 1871-2509 | ||||
Official Date: | 21 March 2013 | ||||
Dates: |
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Volume: | Volume 24 | ||||
Number: | Number 3 | ||||
Page Range: | pp. 371-386 | ||||
DOI: | 10.1090/S1061-0022-2013-01244-1 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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