Signorini problem

The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to its mass forces. The name was coined by Gaetano Fichera to honour his teacher, Antonio Signorini: the original name coined by him is problem with ambiguous boundary conditions.
History.
The problem was posed by Antonio Signorini during a course taught at the "Istituto Nazionale di Alta Matematica" in 1959, later published as the article , expanding a previous short exposition he gave in a note published in 1933. himself called it "problem with ambiguous boundary conditions", since there are two alternative sets of boundary conditions the solution "must satisfy" on any given contact point. The statement of the problem involves not only equalities "but also inequalities", and "it is not a priori known what of the two sets of boundary conditions is satisfied at each point". Signorini asked to determine if the problem is well-posed or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young analysts to study the problem.
Gaetano Fichera and Mauro Picone attended the course, and Fichera started to investigate the problem: since he found no references to similar problems in the theory of boundary value problems, he decided to approach it by starting from first principles, specifically from the virtual work principle.
During Fichera's researches on the problem, Signorini began to suffer serious health problems: nevertheless, he desired to know the answer to his question before his death. Picone, being tied by a strong friendship with Signorini, began to chase Fichera to find a solution: Fichera himself, being tied as well to Signorini by similar feelings, perceived the last months of 1962 as worrying days. Finally, on the first days of January 1963, Fichera was able to give a complete proof of the existence of a unique solution for the problem with ambiguous boundary condition, which he called the "Signorini problem" to honour his teacher. A preliminary research announcement, later published as , was written up and submitted to Signorini exactly a week before his death. Signorini expressed great satisfaction to see a solution to his question.
A few days later, during a conversation with his family Doctor Damiano Aprile, Signorini told him:
According to the solution of the Signorini problem coincides with the birth of the field of variational inequalities.
Formal statement of the problem.
The content of this section and the following subsections follows closely the treatment of Gaetano Fichera in , and also : his derivation of the problem is different from Signorini's one in that he does not consider only incompressible bodies and a plane rest surface, as Signorini does. The problem consist in finding the displacement vector from the natural configuration formula_1 of an anisotropic non-homogeneous elastic body that lies in a subset formula_2 of the three-dimensional euclidean space whose boundary is formula_3 and whose interior normal is the vector formula_4, resting on a rigid frictionless surface whose contact surface (or more generally contact set) is formula_5 and subject only to its body forces formula_6, and surface forces formula_7 applied on the free (i.e. not in contact with the rest surface) surface formula_8: the set formula_2 and the contact surface formula_5 characterize the natural configuration of the body and are known a priori. Therefore, the body has to satisfy the general equilibrium equations
written using the Einstein notation as all in the following development, the ordinary boundary conditions on formula_12
and the following two sets of boundary conditions on formula_5, where formula_15 is the Cauchy stress tensor. Obviously, the body forces and surface forces cannot be given in arbitrary way but they must satisfy a condition in order for the body to reach an equilibrium configuration: this condition will be deduced and analyzed in the following development.
The ambiguous boundary conditions.
If formula_16 is any tangent vector to the contact set formula_5, then the ambiguous boundary condition in each point of this set are expressed by the following two systems of inequalities
Let's analyze their meaning: 
Knowing these facts, the set of conditions applies to points of the boundary of the body which "do not" leave the contact set formula_5 in the equilibrium configuration, since, according to the first relation, the displacement vector formula_27 "has no components" directed as the normal vector formula_4, while, according to the second relation, the tension vector "may have a component" directed as the normal vector formula_4 and having the same sense. In an analogous way, the set of conditions applies to points of the boundary of the body which "leave" that set in the equilibrium configuration, since displacement vector formula_27 "has a component" directed as the normal vector formula_4, while the tension vector "has no components" directed as the normal vector formula_4. For both sets of conditions, the tension vector has no tangent component to the contact set, according to the hypothesis that the body rests on a rigid "frictionless" surface.
Each system expresses a unilateral constraint, in the sense that they express the physical impossibility of the elastic body to penetrate into the surface where it rests: the ambiguity is not only in the unknown values non-zero quantities must satisfy on the contact set but also in the fact that it is not a priori known if a point belonging to that set satisfies the system of boundary conditions or . The set of points where is satisfied is called the area of support of the elastic body on formula_5, while its complement respect to formula_5 is called the area of separation.
The above formulation is "general" since the Cauchy stress tensor i.e. the constitutive equation of the elastic body has not been made explicit: it is equally valid assuming the hypothesis of linear elasticity or the ones of nonlinear elasticity. However, as it would be clear from the following developments, the problem is inherently nonlinear, therefore "assuming a linear stress tensor does not simplify the problem".
The form of the stress tensor in the formulation of Signorini and Fichera.
The form assumed by Signorini and Fichera for the elastic potential energy is the following one (as in the previous developments, the Einstein notation is adopted)
where 
The Cauchy stress tensor has therefore the following form
and it is "linear" with respect to the components of the infinitesimal strain tensor; however, it is not homogeneous nor isotropic.
Solution of the problem.
As for the section on the formal statement of the Signorini problem, the contents of this section and the included subsections follow closely the treatment of Gaetano Fichera in , , and also : obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem , , , and , rather than the technical details.
The potential energy.
The first step of the analysis of Fichera as well as the first step of the analysis of Antonio Signorini in is the analysis of the potential energy, i.e. the following functional
where formula_27 belongs to the set of admissible displacements formula_41 i.e. the set of displacement vectors satisfying the system of boundary conditions or . The meaning of each of the three terms is the following 
 was able to prove that the admissible displacement formula_27 which minimize the integral formula_43 is a solution of the problem with ambiguous boundary conditions , , , and , provided it is a formula_44 function supported on the closure formula_45 of the set formula_2: however Gaetano Fichera gave a class of counterexamples in showing that in general, admissible displacements are 
not smooth functions of these class. Therefore, Fichera tries to minimize the functional in a wider function space: in doing so, he first calculates the first variation (or functional derivative) of the given functional in the neighbourhood of the sought minimizing admissible displacement formula_47, and then requires it to be greater than or equal to zero
Defining the following functionals
and
the preceding inequality is can be written as
This inequality is the variational inequality for the Signorini problem.