Signorini problem

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)