Signorini problem

Given an open subset of with smooth boundary and an function, the Signorini problem consists in finding a function on that is a solution to the following boundary value problem:

Here, is a second-order linear and symmetric elliptic operator on (in particular, can be equal to , the Laplace operator) and is the outward normal derivative to corresponding to . 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 is the field of displacements and is the normal stress (see [a2]). In the Signorini problem, the boundary conditions can be equivalently expressed as:

where is an unknown part of . 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]:

where is the Dirichlet bilinear form associated to and . Here, is the usual Sobolev space on . 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)