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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Optimal regularity and free boundary regularity for the Signorini problem


Author: J. Andersson
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 371-386
MSC (2010): Primary 49J40, 49N60
Published electronically: March 21, 2013
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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.

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Additional Information

J. Andersson
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: j.e.andersson@warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01244-1
PII: S 1061-0022(2013)01244-1
Keywords: Free boundary regularity, Signorini problem, optimal regularity, system of equations
Received by editor(s): November 1, 2011
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society