The problem of finding the minimal value of a real-valued function of real variable

This is a standard example problem, reported by Antman (1983, p. 283): consider the problem of finding the minimal value of a differentiable function ${\displaystyle f}$ over a closed interval ${\displaystyle I=[a,b]}$. Let ${\displaystyle x^{\ast }}$ be a point in ${\displaystyle I}$ where the minimum occurs. Three cases can occur:

1. if ${\displaystyle a<x^{\ast }<b,}$ then ${\displaystyle f^{\prime }(x^{\ast })=0;}$
2. if ${\displaystyle x^{\ast }=a,}$ then ${\displaystyle f^{\prime }(x^{\ast })\geq 0;}$
3. if ${\displaystyle x^{\ast }=b,}$ then ${\displaystyle f^{\prime }(x^{\ast })\leq 0.}$

These necessary conditions can be summarized as the problem of finding ${\displaystyle x^{\ast }\in I}$ such that

${\displaystyle f^{\prime }(x^{\ast })(y-x^{\ast })\geq 0\quad }$ for ${\displaystyle \quad \forall y\in I.}$

The absolute minimum must be searched between the solutions (if more than one) of the preceding inequality: note that the solution is a real number, therefore this is a finite dimensional variational inequality.

The general finite-dimensional variational inequality

A formulation of the general problem in ${\displaystyle \mathbb {R} ^{n}}$ is the following: given a subset ${\displaystyle K}$ of ${\displaystyle \mathbb {R} ^{n}}$ and a mapping ${\displaystyle F\colon K\to \mathbb {R} ^{n}}$, the finite-dimensional variational inequality problem associated with ${\displaystyle K}$ consist of finding a ${\displaystyle n}$-dimensional vector ${\displaystyle x}$ belonging to ${\displaystyle K}$ such that

${\displaystyle \langle F(x),y-x\rangle \geq 0\qquad \forall y\in K}$

where ${\displaystyle \langle \cdot ,\cdot \rangle \colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$ is the standard inner product on the vector space ${\displaystyle \mathbb {R} ^{n}}$.

The variational inequality for the Signorini problem

The classical Signorini problem: what will be the equilibrium configuration of the orange spherically shaped elastic body resting on the blue rigid frictionless plane?

In the historical survey (Fichera 1995), Gaetano Fichera describes the genesis of his solution to the Signorini problem: the problem consist in finding the elastic equilibrium configuration ${\displaystyle {\boldsymbol {u}}({\boldsymbol {x}})=\left(u_{1}({\boldsymbol {x}}),u_{2}({\boldsymbol {x}}),u_{3}({\boldsymbol {x}})\right)}$ of an anisotropic non-homogeneous elastic body that lies in a subset ${\displaystyle A}$ of the three-dimensional euclidean space whose boundary is ${\displaystyle \partial A}$, resting on a rigid frictionless surface and subject only to its mass forces. The solution ${\displaystyle u}$ of the problem exists and is unique (under precise assumptions) in the set of admissible displacements ${\displaystyle {\mathcal {U}}_{\Sigma }}$ i.e. the set of displacement vectors satisfying the system of ambiguous boundary conditions if and only if

${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}}-{\boldsymbol {u}})-F({\boldsymbol {v}}-{\boldsymbol {u}})\geq 0\qquad \forall {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

where ${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})}$ and ${\displaystyle F({\boldsymbol {v}})}$ are the following functionals, written using the Einstein notation

${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})=-\int _{A}\sigma _{ik}({\boldsymbol {u}})\varepsilon _{ik}({\boldsymbol {v}})\,\mathrm {d} x}$,    ${\displaystyle F({\boldsymbol {v}})=\int _{A}v_{i}f_{i}\,\mathrm {d} x+\int _{\partial A\setminus \Sigma }\!\!\!\!\!v_{i}g_{i}\,\mathrm {d} \sigma }$,    ${\displaystyle {\boldsymbol {u}},{\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$

where, for all ${\displaystyle {\boldsymbol {x}}\in A}$,

• ${\displaystyle \Sigma }$ is the contact surface (or more generally a contact set),
• ${\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\left(f_{1}({\boldsymbol {x}}),f_{2}({\boldsymbol {x}}),f_{3}({\boldsymbol {x}})\right)}$ is the body force applied to the body,
• ${\displaystyle {\boldsymbol {g}}({\boldsymbol {x}})=\left(g_{1}({\boldsymbol {x}}),g_{2}({\boldsymbol {x}}),g_{3}({\boldsymbol {x}})\right)}$ is the surface force applied to ${\displaystyle \partial A\!\setminus \!\Sigma }$,
• ${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}({\boldsymbol {u}})=\left(\varepsilon _{ik}({\boldsymbol {u}})\right)=\left({\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}\right)\right)}$ is the infinitesimal strain tensor,
• ${\displaystyle {\boldsymbol {\sigma }}=\left(\sigma _{ik}\right)}$ is the Cauchy stress tensor, defined as

${\displaystyle \sigma _{ik}=-{\frac {\partial W}{\partial \varepsilon _{ik}}}\qquad \forall i,k=1,2,3}$

where ${\displaystyle W({\boldsymbol {\varepsilon }})=a_{ikjh}({\boldsymbol {x}})\varepsilon _{ik}\varepsilon _{jh}}$ is the elastic potential energy and ${\displaystyle {\boldsymbol {a}}({\boldsymbol {x}})=\left(a_{ikjh}({\boldsymbol {x}})\right)}$ is the elasticity tensor.

Historical references

• Antman, Stuart (1983), "The influence of elasticity in analysis: modern developments", Bulletin of the American Mathematical Society, 9 (3): 267–291, doi:10.1090/S0273-0979-1983-15185-6, MR 0714990, Zbl 0533.73001. An historical paper about the fruitful interaction of elasticity theory and mathematical analysis: the creation of the theory of variational inequalities by Gaetano Fichera is described in §5, pages 282–284.
• Duvaut, Georges (1971), "Problèmes unilatéraux en mécanique des milieux continus", Actes du Congrès international des mathématiciens, 1970, ICM Proceedings, vol. Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3, Paris: Gauthier-Villars, pp. 71–78, archived from the original (PDF) on 2015-07-25, retrieved 2015-07-25. A brief research survey describing the field of variational inequalities, precisely the sub-field of continuum mechanics problems with unilateral constraints.
• Fichera, Gaetano (1995), "La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni", Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993, Atti dei Convegni Lincei (in Italian), vol. 114, Roma: Accademia Nazionale dei Lincei, pp. 47–53. The birth of the theory of variational inequalities remembered thirty years later (English translation of the title) is an historical paper describing the beginning of the theory of variational inequalities from the point of view of its founder.

Scientific works

• Facchinei, Francisco; Pang, Jong-Shi (2003), Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1, Springer Series in Operations Research, Berlin–Heidelberg–New York: Springer-Verlag, ISBN 0-387-95580-1, Zbl 1062.90001
• Facchinei, Francisco; Pang, Jong-Shi (2003), Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 2, Springer Series in Operations Research, Berlin–Heidelberg–New York: Springer-Verlag, ISBN 0-387-95581-X, Zbl 1062.90001
• Fichera, Gaetano (1963), "Sul problema elastostatico di Signorini con ambigue condizioni al contorno" [On the elastostatic problem of Signorini with ambiguous boundary conditions], Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 34 (2): 138–142, MR 0176661, Zbl 0128.18305. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
• Fichera, Gaetano (1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno" [Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions], Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 7 (2): 91–140, Zbl 0146.21204. The first paper where an existence and uniqueness theorem for the Signorini problem is proved.
• Fichera, Gaetano (1964b), "Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions", Seminari dell'istituto Nazionale di Alta Matematica 1962–1963, Rome: Edizioni Cremonese, pp. 613–679. An English translation of (Fichera 1964a).
• Glowinski, Roland; Lions, Jacques-Louis; Trémolières, Raymond (1981), Numerical analysis of variational inequalities. Translated from the French, Studies in Mathematics and its Applications, vol. 8, Amsterdam–New York–Oxford: North-Holland, pp. xxix+776, ISBN 0-444-86199-8, MR 0635927, Zbl 0463.65046
• Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, Boston–London–New York–San Diego–Sydney–Tokyo–Toronto: Academic Press, ISBN 0-89871-466-4, Zbl 0457.35001.
• Lions, Jacques-Louis; Stampacchia, Guido (1965), "Inéquations variationnelles non coercives", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 261: 25–27, Zbl 0136.11906, available at Gallica. Announcements of the results of paper (Lions & Stampacchia 1967).
• Lions, Jacques-Louis; Stampacchia, Guido (1967), "Variational inequalities", Communications on Pure and Applied Mathematics, 20 (3): 493–519, doi:10.1002/cpa.3160200302, Zbl 0152.34601, archived from the original on 2013-01-05. An important paper, describing the abstract approach of the authors to the theory of variational inequalities.
• Roubíček, Tomáš (2013), Nonlinear Partial Differential Equations with Applications, ISNM. International Series of Numerical Mathematics, vol. 153 (2nd ed.), Basel–Boston–Berlin: Birkhäuser Verlag, pp. xx+476, doi:10.1007/978-3-0348-0513-1, ISBN 978-3-0348-0512-4, MR 3014456, Zbl 1270.35005.
• Stampacchia, Guido (1964), "Formes bilineaires coercitives sur les ensembles convexes", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 258: 4413–4416, Zbl 0124.06401, available at Gallica. The paper containing Stampacchia's generalization of the Lax–Milgram theorem.