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Given an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101301.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101302.png" /> with smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101304.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101305.png" /> function, the Signorini problem consists in finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101306.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101307.png" /> that is a solution to the following boundary value problem:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101308.png" /></td> </tr></table>
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s1101309.png" /></td> </tr></table>
+
Given an open subset  $  \Omega $
 +
of  $  \mathbf R  ^ {3} $
 +
with smooth boundary  $  \partial  \Omega $
 +
and  $  f $
 +
an  $  L _ {2} ( \Omega ) $
 +
function, the Signorini problem consists in finding a function  $  u $
 +
on  $  \Omega $
 +
that is a solution to the following boundary value problem:
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013010.png" /> is a second-order linear and symmetric elliptic operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013011.png" /> (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013012.png" /> can be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013013.png" />, the [[Laplace operator|Laplace operator]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013014.png" /> is the outward normal derivative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013015.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013016.png" />. This problem, introduced by A. Signorini [[#References|[a5]]] and studied first by G. Fichera [[#References|[a3]]], describes the mathematical model for the deformation of an elastic body whose boundary is in unilateral contact with another elastic body (the static case). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013017.png" /> is the field of displacements and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013018.png" /> is the normal stress (see [[#References|[a2]]]). In the Signorini problem, the boundary conditions can be equivalently expressed as:
+
$$
 +
Au = f  \textrm{ in }  \Omega ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013019.png" /></td> </tr></table>
+
$$
 +
u \geq  0,  {
 +
\frac{\partial  u }{\partial  \nu }
 +
} \geq  0,  u {
 +
\frac{\partial  u }{\partial  \nu }
 +
} = 0 \textrm{ on  }  \partial  \Omega.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013020.png" /></td> </tr></table>
+
Here,  $  A $
 +
is a second-order linear and symmetric elliptic operator on  $  \Omega $(
 +
in particular,  $  A $
 +
can be equal to  $  \Delta $,
 +
the [[Laplace operator|Laplace operator]]) and  $  \partial / {\partial  \nu } $
 +
is the outward normal derivative to  $  \Omega $
 +
corresponding to  $  A $.
 +
This problem, introduced by A. Signorini [[#References|[a5]]] and studied first by G. Fichera [[#References|[a3]]], describes the mathematical model for the deformation of an elastic body whose boundary is in unilateral contact with another elastic body (the static case). In this case  $  u = u ( x ) $
 +
is the field of displacements and  $  {\partial  u } / {\partial  \nu } $
 +
is the normal stress (see [[#References|[a2]]]). In the Signorini problem, the boundary conditions can be equivalently expressed as:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013021.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{\partial  u }{\partial  \nu }
 +
} ,u \geq  0  \textrm{ on  }  \partial  \Omega;
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013022.png" /> is an unknown part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013023.png" />. Thus, the Signorini problem can be viewed as a problem with free boundary, and a weak formulation of the problem is given by the variational inequality [[#References|[a4]]]:
+
$$
 +
u = 0 \textrm{ on  }  \Gamma,  
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013024.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{\partial  u }{\partial  \nu }
 +
} = 0 \textrm{ on  }  \partial  \Omega \setminus  \Gamma,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013025.png" /></td> </tr></table>
+
where  $  \Gamma $
 +
is an unknown part of  $  \partial  \Omega $.
 +
Thus, the Signorini problem can be viewed as a problem with free boundary, and a weak formulation of the problem is given by the variational inequality [[#References|[a4]]]:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013026.png" /> is the Dirichlet bilinear form associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013028.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013029.png" /> is the usual [[Sobolev space|Sobolev space]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110130/s11013030.png" />. In this way the existence and uniqueness of a weak solution to the Signorini problem follows from the general theory of elliptic variational inequalities (see [[#References|[a1]]], [[#References|[a4]]]).
+
$$
 +
u \in K ;
 +
$$
 +
 
 +
$$
 +
a ( u,u - v ) \geq  \int\limits _  \Omega  f ( u - v )  dx,  \forall v \in K \setminus  0,
 +
$$
 +
 
 +
where $  a $
 +
is the Dirichlet bilinear form associated to $  A $
 +
and  $  K = \{ {u \in H  ^ {1} ( \Omega ) } : {u \geq  0 \textrm{ on  }  \partial  \Omega } \} $.  
 +
Here, $  H  ^ {1} ( \Omega ) $
 +
is the usual [[Sobolev space|Sobolev space]] on $  \Omega $.  
 +
In this way the existence and uniqueness of a weak solution to the Signorini problem follows from the general theory of elliptic variational inequalities (see [[#References|[a1]]], [[#References|[a4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brezis,  "Inéquations variationelles"  ''J. Math. Pures Appl.'' , '''51'''  (1972)  pp. 1–168</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Duvaut,  J.L. Lions,  "Inequalities in mechanics and physics" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Fichera,  "Problemi elastostatici con vincoli unilaterali e il problema di Signorini con ambigue condizioni al contorno"  ''Memoirs Acad. Naz. Lincei'' , '''8'''  (1964)  pp. 91–140</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Lions,  G. Stampacchia,  "Variational inequalities"  ''Comm. Pure Appl. Math.'' , '''XX'''  (1967)  pp. 493–519</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Signorini,  "Questioni di elastostatica linearizzata e semilinearizzata"  ''Rend. Mat. Appl.'' , '''XVIII'''  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brezis,  "Inéquations variationelles"  ''J. Math. Pures Appl.'' , '''51'''  (1972)  pp. 1–168</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Duvaut,  J.L. Lions,  "Inequalities in mechanics and physics" , Springer  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Fichera,  "Problemi elastostatici con vincoli unilaterali e il problema di Signorini con ambigue condizioni al contorno"  ''Memoirs Acad. Naz. Lincei'' , '''8'''  (1964)  pp. 91–140</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.L. Lions,  G. Stampacchia,  "Variational inequalities"  ''Comm. Pure Appl. Math.'' , '''XX'''  (1967)  pp. 493–519</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Signorini,  "Questioni di elastostatica linearizzata e semilinearizzata"  ''Rend. Mat. Appl.'' , '''XVIII'''  (1959)</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


Given an open subset $ \Omega $ of $ \mathbf R ^ {3} $ with smooth boundary $ \partial \Omega $ and $ f $ an $ L _ {2} ( \Omega ) $ function, the Signorini problem consists in finding a function $ u $ on $ \Omega $ that is a solution to the following boundary value problem:

$$ Au = f \textrm{ in } \Omega ; $$

$$ u \geq 0, { \frac{\partial u }{\partial \nu } } \geq 0, u { \frac{\partial u }{\partial \nu } } = 0 \textrm{ on } \partial \Omega. $$

Here, $ A $ is a second-order linear and symmetric elliptic operator on $ \Omega $( in particular, $ A $ can be equal to $ \Delta $, the Laplace operator) and $ \partial / {\partial \nu } $ is the outward normal derivative to $ \Omega $ corresponding to $ A $. This problem, introduced by A. Signorini [a5] and studied first by G. Fichera [a3], describes the mathematical model for the deformation of an elastic body whose boundary is in unilateral contact with another elastic body (the static case). In this case $ u = u ( x ) $ is the field of displacements and $ {\partial u } / {\partial \nu } $ is the normal stress (see [a2]). In the Signorini problem, the boundary conditions can be equivalently expressed as:

$$ { \frac{\partial u }{\partial \nu } } ,u \geq 0 \textrm{ on } \partial \Omega; $$

$$ u = 0 \textrm{ on } \Gamma, $$

$$ { \frac{\partial u }{\partial \nu } } = 0 \textrm{ on } \partial \Omega \setminus \Gamma, $$

where $ \Gamma $ is an unknown part of $ \partial \Omega $. Thus, the Signorini problem can be viewed as a problem with free boundary, and a weak formulation of the problem is given by the variational inequality [a4]:

$$ u \in K ; $$

$$ a ( u,u - v ) \geq \int\limits _ \Omega f ( u - v ) dx, \forall v \in K \setminus 0, $$

where $ a $ is the Dirichlet bilinear form associated to $ A $ and $ K = \{ {u \in H ^ {1} ( \Omega ) } : {u \geq 0 \textrm{ on } \partial \Omega } \} $. Here, $ H ^ {1} ( \Omega ) $ is the usual Sobolev space on $ \Omega $. In this way the existence and uniqueness of a weak solution to the Signorini problem follows from the general theory of elliptic variational inequalities (see [a1], [a4]).

References

[a1] H. Brezis, "Inéquations variationelles" J. Math. Pures Appl. , 51 (1972) pp. 1–168
[a2] G. Duvaut, J.L. Lions, "Inequalities in mechanics and physics" , Springer (1976)
[a3] G. Fichera, "Problemi elastostatici con vincoli unilaterali e il problema di Signorini con ambigue condizioni al contorno" Memoirs Acad. Naz. Lincei , 8 (1964) pp. 91–140
[a4] J.L. Lions, G. Stampacchia, "Variational inequalities" Comm. Pure Appl. Math. , XX (1967) pp. 493–519
[a5] A. Signorini, "Questioni di elastostatica linearizzata e semilinearizzata" Rend. Mat. Appl. , XVIII (1959)
How to Cite This Entry:
Signorini problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Signorini_problem&oldid=11805
This article was adapted from an original article by V. Barbu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article