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{{short description|Elastostatics problem in linear elasticity}}
The '''Signorini problem''' is an [[Linear elasticity#Elastostatics|elastostatics]] problem in [[linear elasticity]]: it consists in finding the [[Linear elasticity#Elastostatics|elastic equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of an [[Anisotropy#Material science and engineering|anisotropic]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]], resting on a [[Rigid body|rigid]] [[friction]]less [[Surface (topology)|surface]] and subject only to its [[Weight|mass force]]s. The name was coined by [[Gaetano Fichera]] to honour his teacher, [[Antonio Signorini]]: the original name coined by him is '''problem with ambiguous [[Boundary value problem|boundary conditions]]'''.
== History ==
[[File:Classical Signorini problem.svg|thumb|400px|The classical Signorini problem: what will be the [[Linear elasticity#Elastostatics|equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of the orange spherically shaped [[Physical body|elastic body]] resting on the blue [[Rigid body|rigid]] [[friction]]less [[Plane (geometry)|plane]]?]]
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 {{harv|Signorini|1959}}, expanding a previous short exposition he gave in a note published in 1933. {{harvtxt|Signorini|1959|p=128}} himself called it ''problem with ambiguous [[Boundary value problem|boundary conditions]]'',<ref>{{lang-it|Problema con ambigue condizioni al contorno}}.</ref> since there are two alternative sets of [[Boundary value problem|boundary conditions]] the solution ''must satisfy'' on any given [[Contact (mechanics)|contact point]]. The statement of the problem involves not only [[Equality (mathematics)|equalities]] ''but also [[inequality (mathematics)|inequalities]]'', and ''it is not [[A priori and a posteriori|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 problem|well-posed]] or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young [[mathematical analysis|analysts]] to study the problem.<ref>As it is stated in {{harv|Signorini|1959|p=129}}.</ref>
[[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 problem]]s,<ref>See {{harv|Fichera|1995|p=49}}.</ref> he decided to approach it by starting from [[first principle]]s, 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.<ref>This dramatic situation is described by {{Harvtxt|Fichera|1995|p=51}} himself.</ref> 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 {{Harv|Fichera|1963}}, 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|family Doctor]] Damiano Aprile, Signorini told him:<ref>{{Harvtxt|Fichera|1995|p=53}} reports the episode following the remembraces of [[Mauro Picone]]: see the entry "[[Antonio Signorini]]" for further details.</ref>
{{plainlist|
*"Il mio discepolo Fichera mi ha dato una grande soddisfazione".<ref>{{lang-en|My disciple Fichera gave me a great contentment}}.</ref>
*"Ma Lei ne ha avute tante, Professore, durante la Sua vita",<ref>{{lang-en|But you had many, Professor, during your life}}.</ref> replied Doctor Aprile, but then Signorini replied again:
*"Ma questa è la più grande."<ref>{{lang-en|But this is the greatest one}}.</ref> And those were his last words.
}}
According to {{Harvtxt|Antman|1983|p=282}} the solution of the Signorini problem coincides with the birth of the field of [[Variational inequality|variational inequalities]].
== Formal statement of the problem ==
The content of this section and the following subsections follows closely the treatment of [[Gaetano Fichera]] in {{Harvnb|Fichera|1963}}, {{Harvnb|Fichera|1964b}} and also {{Harvnb|Fichera|1995}}: his derivation of the problem is different from [[Antonio Signorini|Signorini]]'s one in that he does not consider only [[incopressible body|incompressible bodies]] and a plane rest [[Surface (topology)|surface]], as Signorini does.<ref>See {{Harvnb|Signorini|1959|p=127}}) for the original approach.</ref> The problem consist in finding the [[displacement vector]] from the [[Continuum mechanics#Mathematical modeling of a continuum|natural configuration]] <math>\scriptstyle\boldsymbol{u}(\boldsymbol{x})=\left(u_1(\boldsymbol{x}),u_2(\boldsymbol{x}),u_3(\boldsymbol{x})\right)</math> of an [[Anisotropy#Material science and engineering|anisotropic]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]] that lies in a [[subset]] <math>A</math> of the three-[[dimension]]al [[euclidean space]] whose [[boundary (topology)|boundary]] is <math>\scriptstyle\partial A</math> and whose [[interior normal]] is the [[Euclidean vector|vector]] '''''<math>n</math>''''', resting on a [[Rigid body|rigid]] [[frictionless]] [[Surface (topology)|surface]] whose [[Contact (mechanics)|contact]] [[Surface (topology)|surface]] (or more generally contact [[set (mathematics)|set]]) is <math>\Sigma</math> and subject only to its [[body force]]s <math>\scriptstyle\boldsymbol{f}(\boldsymbol{x})=\left(f_1(\boldsymbol{x}),f_2(\boldsymbol{x}),f_3(\boldsymbol{x})\right)</math>, and [[surface force]]s <math>\scriptstyle\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)</math> applied on the free (i.e. not in contact with the rest surface) surface <math>\scriptstyle\partial A\setminus\Sigma </math>: the set <math>A</math> and the contact surface <math>\Sigma</math> characterize the natural configuration of the body and are known a priori. Therefore, the body has to satisfy the general [[Stress (mechanics)#Equilibrium equations and symmetry of the stress tensor|equilibrium equations]]
:{{EquationRef|1|(1){{spaces|5}}}}<math>\qquad\frac{\partial\sigma_{ik}}{\partial x_k}- f_i= 0\qquad\text{for } i=1,2,3</math>
written using the [[Einstein notation]] as all in the following development, the ordinary [[Boundary value problem|boundary conditions]] on <math>\scriptstyle\partial A\setminus\Sigma</math>
:{{EquationRef|2|(2){{spaces|5}}}}<math>\qquad\sigma_{ik}n_k-g_i=0\qquad\text{for } i=1,2,3</math>
and the following two sets of [[Boundary value problem|boundary conditions]] on <math>\Sigma</math>, where '''<math>\scriptstyle\boldsymbol{\sigma} = \boldsymbol{\sigma}(\boldsymbol{u})</math>''' 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 '''''<math>\scriptstyle\boldsymbol{\tau}=(\tau_1,\tau_2,\tau_3)</math>''''' is any [[tangent vector]] to the [[Contact (mechanics)|contact]] [[set (mathematics)|set]] <math>\Sigma</math>, then the ambiguous boundary condition in each [[Point (geometry)|point]] of this set are expressed by the following two systems of [[inequality (mathematics)|inequalities]]
:{{EquationRef|3|(3){{spaces|5}}}}<math>
\quad
\begin{cases}
u_i n_i & = 0 \\
\sigma_{ik} n_i n_k & \geq 0\\
\sigma_{ik} n_i \tau_k & = 0
\end{cases}
</math>{{spaces|5}}or{{spaces|5}}{{EquationRef|4|(4){{spaces|5}}}}<math>
\begin{cases}
u_i n_i & > 0 \\
\sigma_{ik} n_i n_k & = 0 \\
\sigma_{ik} n_i \tau_k & = 0
\end{cases}
</math>
Let's analyze their meaning:
*Each [[set (mathematics)|set]] of conditions consists of three [[Binary relation|relations]], [[Equality (mathematics)|equalities]] or [[inequality (mathematics)|inequalities]], and all the second members are the [[Zero function#Other uses of zero in mathematics|zero function]].
*The [[quantity|quantities]] at first member of each first relation are [[Proportionality (mathematics)|proportional]] to the [[norm (mathematics)|norm]] of the [[vector component|component]] of the [[displacement vector]] directed along the [[normal vector]] '''<math>n</math>'''.
*The quantities at first member of each second relation are proportional to the norm of the component of the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] directed along the [[normal vector]] '''<math>n</math>''',
*The quantities at the first member of each third relation are proportional to the norm of the component of the tension vector along any [[Euclidean vector|vector]] '''<math>\tau</math>''' [[Tangent vector|tangent]] in the given [[Point (geometry)|point]] to the [[Contact (mechanics)|contact]] [[set (mathematics)|set]] <math>\Sigma</math>.
*The quantities at the first member of each of the three relations are [[Positive number|positive]] if they have the same [[Euclidean vector|sense]] of the [[Euclidean vector|vector]] they are [[Proportionality (mathematics)|proportional]] to, while they are [[negative number|negative]] if not, therefore the [[Proportionality (mathematics)|constants of proportionality]] are respectively <math>\scriptstyle +1</math> and <math>\scriptstyle -1</math>.
Knowing these facts, the set of conditions {{EquationNote|3|(3)}} applies to [[Point (geometry)|point]]s of the [[boundary (topology)|boundary]] of the body which ''do not'' leave the [[Contact (mechanics)|contact]] set <math>\Sigma</math> in the [[Linear elasticity#Elastostatics|equilibrium configuration]], since, according to the first [[Binary relation|relation]], the [[displacement vector]] '''<math>u</math>''' ''has no [[vector component|component]]s'' directed as the [[normal vector]] '''<math>n</math>''', while, according to the second relation, the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] ''may have a component'' directed as the normal vector '''<math>n</math>''' and having the same [[Euclidean vector|sense]]. In an analogous way, the set of conditions {{EquationNote|4|(4)}} applies to points of the boundary of the body which ''leave'' that set in the equilibrium configuration, since displacement vector '''<math>u</math>''' ''has a component'' directed as the normal vector '''<math>n</math>''', while the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] ''has no components'' directed as the normal vector '''<math>n</math>'''. For both sets of conditions, the tension vector has no tangent component to the [[Contact (mechanics)|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 [[Physical body|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 (mechanics)|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 {{EquationNote|3|(3)}} or {{EquationNote|4|(4)}}. The set of points where {{EquationNote|3|(3)}} is satisfied is called the '''area of support''' of the elastic body on <math>\Sigma</math>, while its [[Complement (set theory)#Relative complement|complement respect to <math>\Sigma</math>]] is called the '''area of separation'''.
The above formulation is ''general'' since the [[Cauchy stress tensor]] i.e. the [[constitutive equation]] of the [[Physical body|elastic body]] has not been made explicit: it is equally valid assuming the [[hypothesis]] of [[linear elasticity]] or the ones of [[Finite strain theory|nonlinear elasticity]]. However, as it would be clear from the following developments, the problem is inherently [[Nonlinear system|nonlinear]], therefore ''assuming a [[Linear elasticity#Mathematical formulation|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 [[Antonio Signorini|Signorini]] and [[Gaetano Fichera|Fichera]] for the [[elastic potential energy]] is the following one (as in the previous developments, the [[Einstein notation]] is adopted)
:<math>W(\boldsymbol{\varepsilon})=a_{ik,jh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{jh}</math>
where
*<math>\scriptstyle\boldsymbol{a}(\boldsymbol{x})=\left(a_{ik,jh}(\boldsymbol{x})\right)</math> is the [[elasticity tensor]]
*<math>\scriptstyle\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)</math> is the [[Infinitesimal strain#Infinitesimal strain tensor|infinitesimal strain tensor]]
The [[Cauchy stress tensor]] has therefore the following form
:{{EquationRef|5|(5){{spaces|5}}}}<math>\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\text{for } i,k=1,2,3</math>
and it is ''[[Linear map|linear]]'' with respect to the components of the infinitesimal strain tensor; however, it is not [[Homogeneity (physics)|homogeneous]] nor [[Isotropy|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 {{Harvnb|Fichera|1963}}, {{Harvnb|Fichera|1964b}}, {{Harvnb|Fichera|1972}} and also {{Harvnb|Fichera|1995}}: obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem {{EquationNote|1|(1)}}, {{EquationNote|2|(2)}}, {{EquationNote|3|(3)}}, {{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, 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 {{Harvnb|Signorini|1959|}} is the analysis of the '''potential energy''', i.e. the following [[Functional (mathematics)|functional]]
:{{EquationRef|6|(6){{spaces|6}}}}<math>I(\boldsymbol{u})=\int_A W(\boldsymbol{x},\boldsymbol{\varepsilon})\mathrm{d}x - \int_A u_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}u_i g_i \mathrm{d}\sigma</math>
where '''<math>u</math>''' belongs to the [[set (mathematics)|set]] of '''admissible displacements''' <math>\scriptstyle\mathcal{U}_\Sigma</math> i.e. the set of [[displacement vector]]s satisfying the system of [[boundary value problem|boundary conditions]] {{EquationNote|3|(3)}} or {{EquationNote|4|(4)}}. The meaning of each of the three terms is the following
*the first one is the total [[elastic potential energy]] of the [[Physical body|elastic body]]
*the second one is the total [[potential energy]] due to the [[body force]]s, for example the [[gravitational force]]
*the third one is the potential energy due to [[surface force]]s, for example the [[Force (physics)|force]]s exerted by the [[atmospheric pressure]]
{{Harvtxt|Signorini|1959|pp=129–133}} was able to prove that the admissible displacement '''<math>u</math>''' which [[minimum|minimize]] the integral '''<math>I(u)</math>''' is a solution of the problem with ambiguous boundary conditions {{EquationNote|1|(1)}}, {{EquationNote|2|(2)}}, {{EquationNote|3|(3)}}, {{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, provided it is a [[smooth function|<math>C^1</math> function]] [[Distribution (mathematics)#Support of a distribution|supported]] on the [[Closure (topology)|closure]] <math>\scriptstyle \bar A</math> of the set <math>A</math>: however [[Gaetano Fichera]] gave a class of [[counterexample]]s in {{Harv|Fichera|1964b|pp=619–620}} showing that in general, admissible displacements are
not [[smooth function]]s of these class. Therefore, Fichera tries to minimize the [[Functional (mathematics)|functional]] {{EquationNote|6|(6)}} in a wider [[function space]]: in doing so, he first calculates the [[first variation]] (or [[functional derivative]]) of the given functional in the [[Neighbourhood (topology)|neighbourhood]] of the sought minimizing admissible displacement <math>\scriptstyle\boldsymbol{u} \in \mathcal{U}_\Sigma</math>, and then requires it to be greater than or equal to [[zero]]
:<math>\left. \frac{\mathrm{d}}{\mathrm{d}t} I( \boldsymbol{u} + t \boldsymbol{v}) \right\vert_{t=0} = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x - \int_A v_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma</math>
Defining the following functionals
:<math>B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x \qquad \boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma</math>
and
:<math>F(\boldsymbol{v}) = \int_A v_i f_i\mathrm{d}x + \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma\qquad \boldsymbol{v} \in \mathcal{U}_\Sigma</math>
the preceding [[inequality (mathematics)|inequality]] is can be written as
:{{EquationRef|7|(7){{spaces|6}}}}<math>B(\boldsymbol{u},\boldsymbol{v}) - F(\boldsymbol{v}) \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma </math>
This inequality is the '''[[variational inequality]] for the Signorini problem'''.
== See also ==
* [[Linear elasticity]]
* [[Variational inequality]]
== Notes ==
{{Reflist|30em}}
==References==
=== Historical references ===
*{{Citation
| last = Antman
| first = Stuart | authorlink = Stuart Antman
| title = The influence of elasticity in analysis: modern developments
| journal = [[Bulletin of the American Mathematical Society]]
| volume = 9
| issue = 3
| pages = 267–291
| year = 1983
| url = http://www.ams.org/bull/1983-09-03/S0273-0979-1983-15185-6/home.html
| doi = 10.1090/S0273-0979-1983-15185-6
| mr = 714990
| zbl = 0533.73001
| doi-access = free
}}.
*{{Citation
| last = Duvaut
| first = Georges
| author-link =Georges Duvaut
| contribution = Problèmes unilatéraux en mécanique des milieux continus
| contribution-url = http://www.mathunion.org/ICM/ICM1970.3/Main/icm1970.3.0071.0078.ocr.pdf
| series = [[International Congress of Mathematicians|ICM Proceedings]]
| title = Actes du Congrès international des mathématiciens, 1970
| volume = Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3
| pages = 71–78
| year = 1971
| place = [[Paris]]
| publisher = [[Gauthier-Villars]]
| url = http://www.mathunion.org/ICM/ICM1970.3/
| id =
| mr =
| zbl =
}}. A brief research survey describing the field.
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| contribution = Boundary value problems of elasticity with unilateral constraints
| year = 1972
| title = Festkörpermechanik/Mechanics of Solids
| editor-last = Flügge
| editor-first = Siegfried
| editor-link = Siegfried Flügge
| editor2-last = Truesdell
| editor2-first = Clifford A.
| editor2-link = Clifford Truesdell
| series = Handbuch der Physik (Encyclopedia of Physics)
| volume = VIa/2
| edition = paperback 1984
| pages = 391–424
| place = Berlin–[[Heidelberg]]–New York
| publisher = [[Springer-Verlag]]
| zbl = 0277.73001
| isbn = 0-387-13161-2
}}. The encyclopedia entry about problems with unilateral constraints (the class of [[boundary value problem]]s the Signorini problem belongs to) he wrote for the ''Handbuch der Physik'' on invitation by [[Clifford Truesdell]].
*{{Citation
| first = Gaetano
| last = Fichera
| author-link = Gaetano Fichera
| editor-last =
| editor-first =
| contribution = La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni
| title = Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
| url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32885
| language = Italian
| year = 1995
| pages = 47–53
| place = [[Rome|Roma]]
| series = Atti dei Convegni Lincei
| volume = 114
| publisher = [[Accademia Nazionale dei Lincei]]
}}. ''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.
* {{citation
| last = Fichera
| first = Gaetano
| title = Opere storiche biografiche, divulgative
| publisher = Giannini
| location = [[Napoli]]
| year = 2002
| language = Italian
| pages = 491
}}. "''Historical, biographical, divulgative works''" in the English translation: a volume collecting almost all works of Gaetano Fichera in the fields of [[history of mathematics]] and scientific divulgation.
* {{citation
|last = Fichera
|first = Gaetano
|title = Opere scelte
|url = http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|location = [[Firenze]]
|year = 2004
|pages = XXIX+432 (vol. 1), pp. VI+570 (vol. 2), pp. VI+583 (vol. 3)
|url-status = dead
|archiveurl = https://web.archive.org/web/20091228075048/http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|archivedate = 2009-12-28
}}, {{isbn|88-7083-811-0}} (vol. 1), {{isbn|88-7083-812-9}} (vol. 2), {{isbn|88-7083-813-7}} (vol. 3). Gaetano Fichera's "''Selected works''": three volumes collecting his most important mathematical papers, with a biographical sketch of [[Olga Arsenievna Oleinik|Olga A. Oleinik]].
* {{citation
|last = Signorini
|first = Antonio
|author-link = Antonio Signorini
|title = Opere scelte
|url = http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|location = [[Firenze]]
|year = 1991
|pages = XXXI + 695
|url-status = dead
|archiveurl = https://web.archive.org/web/20091228075048/http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|archivedate = 2009-12-28
}}. The "''Selected works''" of Antonio Signorini: a volume collecting his most important works with an introduction and a commentary of [[Giuseppe Grioli]].
===Research works===
*{{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Sul problema elastostatico di Signorini con ambigue condizioni al contorno
| journal = Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 34
| series = 8
| issue = 2
| year = 1963
| pages=138–142
| zbl= 0128.18305
}}. "''On the elastostatic problem of Signorini with ambiguous boundary conditions''" (English translation of the title) is a short research note announcing and describing the solution of the Signorini problem.
*{{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno
| journal = Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 7
| series = 8
| issue = 2
| year = 1964a
| pages=91–140
| zbl = 0146.21204
}}. "''Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions''" (English translation of the title) is the first paper where aa [[Existence theorem|existence]] and [[uniqueness theorem]] for the Signorini problem is proved.
* {{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| contribution = Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions
| title = Seminari dell'istituto Nazionale di Alta Matematica 1962–1963
| year = 1964b
| publisher = Edizioni Cremonese
| place = [[Rome]]
| pages=613–679
}}. An English translation of the previous paper.
*{{Citation
| last = Signorini
| first = Antonio
| author-link = Antonio Signorini
| title = Questioni di elasticità non linearizzata e semilinearizzata
|trans-title=Topics in non linear and semilinear elasticity
| journal = [[Rendiconti di Matematica e delle sue Applicazioni]]
| language = Italian
| series = 5
| volume = 18
| pages = 95–139
| year = 1959
| zbl = 0091.38006
}}.
*{{citation|last=Petrosyan|first=Arshak|last2=Shahgholian|first2=Henrik|last3=Uraltseva|first3=Nina|title=Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics|publisher=American Mathematical Society, Providence, RI|year=2012|isbn=978-0-8218-8794-3 }}.
*{{Citation
| last = Andersson
| first = John
| title = Optimal regularity for the Signorini problem and its free boundary
| journal = [[Invent. Math.]]
| language = English
| series =
| volume = 1
| pages = 1–82
| year = 2016
| issue = 1
| doi = 10.1007/s00222-015-0608-6
| arxiv = 1310.2511
| bibcode = 2016InMat.204....1A
| zbl =
}}.
== External links ==
*{{springer
| title= Signorini problem
| id= S/s110130
| last= Barbu
| first= V.
}}
* [https://www.scilag.net/problem/G-180630.1 Alessio Figalli, On global homogeneous solutions to the Signorini problem],
{{DEFAULTSORT:Signorini Problem}}
[[Category:Calculus of variations]]
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Partial differential equations]]
The '''Signorini problem''' is an [[Linear elasticity#Elastostatics|elastostatics]] problem in [[linear elasticity]]: it consists in finding the [[Linear elasticity#Elastostatics|elastic equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of an [[Anisotropy#Material science and engineering|anisotropic]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]], resting on a [[Rigid body|rigid]] [[friction]]less [[Surface (topology)|surface]] and subject only to its [[Weight|mass force]]s. The name was coined by [[Gaetano Fichera]] to honour his teacher, [[Antonio Signorini]]: the original name coined by him is '''problem with ambiguous [[Boundary value problem|boundary conditions]]'''.
== History ==
[[File:Classical Signorini problem.svg|thumb|400px|The classical Signorini problem: what will be the [[Linear elasticity#Elastostatics|equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of the orange spherically shaped [[Physical body|elastic body]] resting on the blue [[Rigid body|rigid]] [[friction]]less [[Plane (geometry)|plane]]?]]
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 {{harv|Signorini|1959}}, expanding a previous short exposition he gave in a note published in 1933. {{harvtxt|Signorini|1959|p=128}} himself called it ''problem with ambiguous [[Boundary value problem|boundary conditions]]'',<ref>{{lang-it|Problema con ambigue condizioni al contorno}}.</ref> since there are two alternative sets of [[Boundary value problem|boundary conditions]] the solution ''must satisfy'' on any given [[Contact (mechanics)|contact point]]. The statement of the problem involves not only [[Equality (mathematics)|equalities]] ''but also [[inequality (mathematics)|inequalities]]'', and ''it is not [[A priori and a posteriori|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 problem|well-posed]] or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young [[mathematical analysis|analysts]] to study the problem.<ref>As it is stated in {{harv|Signorini|1959|p=129}}.</ref>
[[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 problem]]s,<ref>See {{harv|Fichera|1995|p=49}}.</ref> he decided to approach it by starting from [[first principle]]s, 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.<ref>This dramatic situation is described by {{Harvtxt|Fichera|1995|p=51}} himself.</ref> 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 {{Harv|Fichera|1963}}, 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|family Doctor]] Damiano Aprile, Signorini told him:<ref>{{Harvtxt|Fichera|1995|p=53}} reports the episode following the remembraces of [[Mauro Picone]]: see the entry "[[Antonio Signorini]]" for further details.</ref>
{{plainlist|
*"Il mio discepolo Fichera mi ha dato una grande soddisfazione".<ref>{{lang-en|My disciple Fichera gave me a great contentment}}.</ref>
*"Ma Lei ne ha avute tante, Professore, durante la Sua vita",<ref>{{lang-en|But you had many, Professor, during your life}}.</ref> replied Doctor Aprile, but then Signorini replied again:
*"Ma questa è la più grande."<ref>{{lang-en|But this is the greatest one}}.</ref> And those were his last words.
}}
According to {{Harvtxt|Antman|1983|p=282}} the solution of the Signorini problem coincides with the birth of the field of [[Variational inequality|variational inequalities]].
== Formal statement of the problem ==
The content of this section and the following subsections follows closely the treatment of [[Gaetano Fichera]] in {{Harvnb|Fichera|1963}}, {{Harvnb|Fichera|1964b}} and also {{Harvnb|Fichera|1995}}: his derivation of the problem is different from [[Antonio Signorini|Signorini]]'s one in that he does not consider only [[incopressible body|incompressible bodies]] and a plane rest [[Surface (topology)|surface]], as Signorini does.<ref>See {{Harvnb|Signorini|1959|p=127}}) for the original approach.</ref> The problem consist in finding the [[displacement vector]] from the [[Continuum mechanics#Mathematical modeling of a continuum|natural configuration]] <math>\scriptstyle\boldsymbol{u}(\boldsymbol{x})=\left(u_1(\boldsymbol{x}),u_2(\boldsymbol{x}),u_3(\boldsymbol{x})\right)</math> of an [[Anisotropy#Material science and engineering|anisotropic]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]] that lies in a [[subset]] <math>A</math> of the three-[[dimension]]al [[euclidean space]] whose [[boundary (topology)|boundary]] is <math>\scriptstyle\partial A</math> and whose [[interior normal]] is the [[Euclidean vector|vector]] '''''<math>n</math>''''', resting on a [[Rigid body|rigid]] [[frictionless]] [[Surface (topology)|surface]] whose [[Contact (mechanics)|contact]] [[Surface (topology)|surface]] (or more generally contact [[set (mathematics)|set]]) is <math>\Sigma</math> and subject only to its [[body force]]s <math>\scriptstyle\boldsymbol{f}(\boldsymbol{x})=\left(f_1(\boldsymbol{x}),f_2(\boldsymbol{x}),f_3(\boldsymbol{x})\right)</math>, and [[surface force]]s <math>\scriptstyle\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)</math> applied on the free (i.e. not in contact with the rest surface) surface <math>\scriptstyle\partial A\setminus\Sigma </math>: the set <math>A</math> and the contact surface <math>\Sigma</math> characterize the natural configuration of the body and are known a priori. Therefore, the body has to satisfy the general [[Stress (mechanics)#Equilibrium equations and symmetry of the stress tensor|equilibrium equations]]
:{{EquationRef|1|(1){{spaces|5}}}}<math>\qquad\frac{\partial\sigma_{ik}}{\partial x_k}- f_i= 0\qquad\text{for } i=1,2,3</math>
written using the [[Einstein notation]] as all in the following development, the ordinary [[Boundary value problem|boundary conditions]] on <math>\scriptstyle\partial A\setminus\Sigma</math>
:{{EquationRef|2|(2){{spaces|5}}}}<math>\qquad\sigma_{ik}n_k-g_i=0\qquad\text{for } i=1,2,3</math>
and the following two sets of [[Boundary value problem|boundary conditions]] on <math>\Sigma</math>, where '''<math>\scriptstyle\boldsymbol{\sigma} = \boldsymbol{\sigma}(\boldsymbol{u})</math>''' 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 '''''<math>\scriptstyle\boldsymbol{\tau}=(\tau_1,\tau_2,\tau_3)</math>''''' is any [[tangent vector]] to the [[Contact (mechanics)|contact]] [[set (mathematics)|set]] <math>\Sigma</math>, then the ambiguous boundary condition in each [[Point (geometry)|point]] of this set are expressed by the following two systems of [[inequality (mathematics)|inequalities]]
:{{EquationRef|3|(3){{spaces|5}}}}<math>
\quad
\begin{cases}
u_i n_i & = 0 \\
\sigma_{ik} n_i n_k & \geq 0\\
\sigma_{ik} n_i \tau_k & = 0
\end{cases}
</math>{{spaces|5}}or{{spaces|5}}{{EquationRef|4|(4){{spaces|5}}}}<math>
\begin{cases}
u_i n_i & > 0 \\
\sigma_{ik} n_i n_k & = 0 \\
\sigma_{ik} n_i \tau_k & = 0
\end{cases}
</math>
Let's analyze their meaning:
*Each [[set (mathematics)|set]] of conditions consists of three [[Binary relation|relations]], [[Equality (mathematics)|equalities]] or [[inequality (mathematics)|inequalities]], and all the second members are the [[Zero function#Other uses of zero in mathematics|zero function]].
*The [[quantity|quantities]] at first member of each first relation are [[Proportionality (mathematics)|proportional]] to the [[norm (mathematics)|norm]] of the [[vector component|component]] of the [[displacement vector]] directed along the [[normal vector]] '''<math>n</math>'''.
*The quantities at first member of each second relation are proportional to the norm of the component of the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] directed along the [[normal vector]] '''<math>n</math>''',
*The quantities at the first member of each third relation are proportional to the norm of the component of the tension vector along any [[Euclidean vector|vector]] '''<math>\tau</math>''' [[Tangent vector|tangent]] in the given [[Point (geometry)|point]] to the [[Contact (mechanics)|contact]] [[set (mathematics)|set]] <math>\Sigma</math>.
*The quantities at the first member of each of the three relations are [[Positive number|positive]] if they have the same [[Euclidean vector|sense]] of the [[Euclidean vector|vector]] they are [[Proportionality (mathematics)|proportional]] to, while they are [[negative number|negative]] if not, therefore the [[Proportionality (mathematics)|constants of proportionality]] are respectively <math>\scriptstyle +1</math> and <math>\scriptstyle -1</math>.
Knowing these facts, the set of conditions {{EquationNote|3|(3)}} applies to [[Point (geometry)|point]]s of the [[boundary (topology)|boundary]] of the body which ''do not'' leave the [[Contact (mechanics)|contact]] set <math>\Sigma</math> in the [[Linear elasticity#Elastostatics|equilibrium configuration]], since, according to the first [[Binary relation|relation]], the [[displacement vector]] '''<math>u</math>''' ''has no [[vector component|component]]s'' directed as the [[normal vector]] '''<math>n</math>''', while, according to the second relation, the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] ''may have a component'' directed as the normal vector '''<math>n</math>''' and having the same [[Euclidean vector|sense]]. In an analogous way, the set of conditions {{EquationNote|4|(4)}} applies to points of the boundary of the body which ''leave'' that set in the equilibrium configuration, since displacement vector '''<math>u</math>''' ''has a component'' directed as the normal vector '''<math>n</math>''', while the [[Stress (mechanics)#Relationship stress vector - stress tensor|tension vector]] ''has no components'' directed as the normal vector '''<math>n</math>'''. For both sets of conditions, the tension vector has no tangent component to the [[Contact (mechanics)|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 [[Physical body|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 (mechanics)|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 {{EquationNote|3|(3)}} or {{EquationNote|4|(4)}}. The set of points where {{EquationNote|3|(3)}} is satisfied is called the '''area of support''' of the elastic body on <math>\Sigma</math>, while its [[Complement (set theory)#Relative complement|complement respect to <math>\Sigma</math>]] is called the '''area of separation'''.
The above formulation is ''general'' since the [[Cauchy stress tensor]] i.e. the [[constitutive equation]] of the [[Physical body|elastic body]] has not been made explicit: it is equally valid assuming the [[hypothesis]] of [[linear elasticity]] or the ones of [[Finite strain theory|nonlinear elasticity]]. However, as it would be clear from the following developments, the problem is inherently [[Nonlinear system|nonlinear]], therefore ''assuming a [[Linear elasticity#Mathematical formulation|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 [[Antonio Signorini|Signorini]] and [[Gaetano Fichera|Fichera]] for the [[elastic potential energy]] is the following one (as in the previous developments, the [[Einstein notation]] is adopted)
:<math>W(\boldsymbol{\varepsilon})=a_{ik,jh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{jh}</math>
where
*<math>\scriptstyle\boldsymbol{a}(\boldsymbol{x})=\left(a_{ik,jh}(\boldsymbol{x})\right)</math> is the [[elasticity tensor]]
*<math>\scriptstyle\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)</math> is the [[Infinitesimal strain#Infinitesimal strain tensor|infinitesimal strain tensor]]
The [[Cauchy stress tensor]] has therefore the following form
:{{EquationRef|5|(5){{spaces|5}}}}<math>\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\text{for } i,k=1,2,3</math>
and it is ''[[Linear map|linear]]'' with respect to the components of the infinitesimal strain tensor; however, it is not [[Homogeneity (physics)|homogeneous]] nor [[Isotropy|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 {{Harvnb|Fichera|1963}}, {{Harvnb|Fichera|1964b}}, {{Harvnb|Fichera|1972}} and also {{Harvnb|Fichera|1995}}: obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem {{EquationNote|1|(1)}}, {{EquationNote|2|(2)}}, {{EquationNote|3|(3)}}, {{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, 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 {{Harvnb|Signorini|1959|}} is the analysis of the '''potential energy''', i.e. the following [[Functional (mathematics)|functional]]
:{{EquationRef|6|(6){{spaces|6}}}}<math>I(\boldsymbol{u})=\int_A W(\boldsymbol{x},\boldsymbol{\varepsilon})\mathrm{d}x - \int_A u_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}u_i g_i \mathrm{d}\sigma</math>
where '''<math>u</math>''' belongs to the [[set (mathematics)|set]] of '''admissible displacements''' <math>\scriptstyle\mathcal{U}_\Sigma</math> i.e. the set of [[displacement vector]]s satisfying the system of [[boundary value problem|boundary conditions]] {{EquationNote|3|(3)}} or {{EquationNote|4|(4)}}. The meaning of each of the three terms is the following
*the first one is the total [[elastic potential energy]] of the [[Physical body|elastic body]]
*the second one is the total [[potential energy]] due to the [[body force]]s, for example the [[gravitational force]]
*the third one is the potential energy due to [[surface force]]s, for example the [[Force (physics)|force]]s exerted by the [[atmospheric pressure]]
{{Harvtxt|Signorini|1959|pp=129–133}} was able to prove that the admissible displacement '''<math>u</math>''' which [[minimum|minimize]] the integral '''<math>I(u)</math>''' is a solution of the problem with ambiguous boundary conditions {{EquationNote|1|(1)}}, {{EquationNote|2|(2)}}, {{EquationNote|3|(3)}}, {{EquationNote|4|(4)}} and {{EquationNote|5|(5)}}, provided it is a [[smooth function|<math>C^1</math> function]] [[Distribution (mathematics)#Support of a distribution|supported]] on the [[Closure (topology)|closure]] <math>\scriptstyle \bar A</math> of the set <math>A</math>: however [[Gaetano Fichera]] gave a class of [[counterexample]]s in {{Harv|Fichera|1964b|pp=619–620}} showing that in general, admissible displacements are
not [[smooth function]]s of these class. Therefore, Fichera tries to minimize the [[Functional (mathematics)|functional]] {{EquationNote|6|(6)}} in a wider [[function space]]: in doing so, he first calculates the [[first variation]] (or [[functional derivative]]) of the given functional in the [[Neighbourhood (topology)|neighbourhood]] of the sought minimizing admissible displacement <math>\scriptstyle\boldsymbol{u} \in \mathcal{U}_\Sigma</math>, and then requires it to be greater than or equal to [[zero]]
:<math>\left. \frac{\mathrm{d}}{\mathrm{d}t} I( \boldsymbol{u} + t \boldsymbol{v}) \right\vert_{t=0} = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x - \int_A v_i f_i\mathrm{d}x - \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma</math>
Defining the following functionals
:<math>B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x \qquad \boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma</math>
and
:<math>F(\boldsymbol{v}) = \int_A v_i f_i\mathrm{d}x + \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \mathrm{d}\sigma\qquad \boldsymbol{v} \in \mathcal{U}_\Sigma</math>
the preceding [[inequality (mathematics)|inequality]] is can be written as
:{{EquationRef|7|(7){{spaces|6}}}}<math>B(\boldsymbol{u},\boldsymbol{v}) - F(\boldsymbol{v}) \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma </math>
This inequality is the '''[[variational inequality]] for the Signorini problem'''.
== See also ==
* [[Linear elasticity]]
* [[Variational inequality]]
== Notes ==
{{Reflist|30em}}
==References==
=== Historical references ===
*{{Citation
| last = Antman
| first = Stuart | authorlink = Stuart Antman
| title = The influence of elasticity in analysis: modern developments
| journal = [[Bulletin of the American Mathematical Society]]
| volume = 9
| issue = 3
| pages = 267–291
| year = 1983
| url = http://www.ams.org/bull/1983-09-03/S0273-0979-1983-15185-6/home.html
| doi = 10.1090/S0273-0979-1983-15185-6
| mr = 714990
| zbl = 0533.73001
| doi-access = free
}}.
*{{Citation
| last = Duvaut
| first = Georges
| author-link =Georges Duvaut
| contribution = Problèmes unilatéraux en mécanique des milieux continus
| contribution-url = http://www.mathunion.org/ICM/ICM1970.3/Main/icm1970.3.0071.0078.ocr.pdf
| series = [[International Congress of Mathematicians|ICM Proceedings]]
| title = Actes du Congrès international des mathématiciens, 1970
| volume = Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3
| pages = 71–78
| year = 1971
| place = [[Paris]]
| publisher = [[Gauthier-Villars]]
| url = http://www.mathunion.org/ICM/ICM1970.3/
| id =
| mr =
| zbl =
}}. A brief research survey describing the field.
*{{Citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| contribution = Boundary value problems of elasticity with unilateral constraints
| year = 1972
| title = Festkörpermechanik/Mechanics of Solids
| editor-last = Flügge
| editor-first = Siegfried
| editor-link = Siegfried Flügge
| editor2-last = Truesdell
| editor2-first = Clifford A.
| editor2-link = Clifford Truesdell
| series = Handbuch der Physik (Encyclopedia of Physics)
| volume = VIa/2
| edition = paperback 1984
| pages = 391–424
| place = Berlin–[[Heidelberg]]–New York
| publisher = [[Springer-Verlag]]
| zbl = 0277.73001
| isbn = 0-387-13161-2
}}. The encyclopedia entry about problems with unilateral constraints (the class of [[boundary value problem]]s the Signorini problem belongs to) he wrote for the ''Handbuch der Physik'' on invitation by [[Clifford Truesdell]].
*{{Citation
| first = Gaetano
| last = Fichera
| author-link = Gaetano Fichera
| editor-last =
| editor-first =
| contribution = La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni
| title = Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
| url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32885
| language = Italian
| year = 1995
| pages = 47–53
| place = [[Rome|Roma]]
| series = Atti dei Convegni Lincei
| volume = 114
| publisher = [[Accademia Nazionale dei Lincei]]
}}. ''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.
* {{citation
| last = Fichera
| first = Gaetano
| title = Opere storiche biografiche, divulgative
| publisher = Giannini
| location = [[Napoli]]
| year = 2002
| language = Italian
| pages = 491
}}. "''Historical, biographical, divulgative works''" in the English translation: a volume collecting almost all works of Gaetano Fichera in the fields of [[history of mathematics]] and scientific divulgation.
* {{citation
|last = Fichera
|first = Gaetano
|title = Opere scelte
|url = http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|location = [[Firenze]]
|year = 2004
|pages = XXIX+432 (vol. 1), pp. VI+570 (vol. 2), pp. VI+583 (vol. 3)
|url-status = dead
|archiveurl = https://web.archive.org/web/20091228075048/http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|archivedate = 2009-12-28
}}, {{isbn|88-7083-811-0}} (vol. 1), {{isbn|88-7083-812-9}} (vol. 2), {{isbn|88-7083-813-7}} (vol. 3). Gaetano Fichera's "''Selected works''": three volumes collecting his most important mathematical papers, with a biographical sketch of [[Olga Arsenievna Oleinik|Olga A. Oleinik]].
* {{citation
|last = Signorini
|first = Antonio
|author-link = Antonio Signorini
|title = Opere scelte
|url = http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|location = [[Firenze]]
|year = 1991
|pages = XXXI + 695
|url-status = dead
|archiveurl = https://web.archive.org/web/20091228075048/http://umi.dm.unibo.it/italiano/Editoria/libri/ogm.html
|archivedate = 2009-12-28
}}. The "''Selected works''" of Antonio Signorini: a volume collecting his most important works with an introduction and a commentary of [[Giuseppe Grioli]].
===Research works===
*{{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Sul problema elastostatico di Signorini con ambigue condizioni al contorno
| journal = Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 34
| series = 8
| issue = 2
| year = 1963
| pages=138–142
| zbl= 0128.18305
}}. "''On the elastostatic problem of Signorini with ambiguous boundary conditions''" (English translation of the title) is a short research note announcing and describing the solution of the Signorini problem.
*{{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| title = Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno
| journal = Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 7
| series = 8
| issue = 2
| year = 1964a
| pages=91–140
| zbl = 0146.21204
}}. "''Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions''" (English translation of the title) is the first paper where aa [[Existence theorem|existence]] and [[uniqueness theorem]] for the Signorini problem is proved.
* {{citation
| last = Fichera
| first = Gaetano
| author-link = Gaetano Fichera
| contribution = Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions
| title = Seminari dell'istituto Nazionale di Alta Matematica 1962–1963
| year = 1964b
| publisher = Edizioni Cremonese
| place = [[Rome]]
| pages=613–679
}}. An English translation of the previous paper.
*{{Citation
| last = Signorini
| first = Antonio
| author-link = Antonio Signorini
| title = Questioni di elasticità non linearizzata e semilinearizzata
|trans-title=Topics in non linear and semilinear elasticity
| journal = [[Rendiconti di Matematica e delle sue Applicazioni]]
| language = Italian
| series = 5
| volume = 18
| pages = 95–139
| year = 1959
| zbl = 0091.38006
}}.
*{{citation|last=Petrosyan|first=Arshak|last2=Shahgholian|first2=Henrik|last3=Uraltseva|first3=Nina|title=Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics|publisher=American Mathematical Society, Providence, RI|year=2012|isbn=978-0-8218-8794-3 }}.
*{{Citation
| last = Andersson
| first = John
| title = Optimal regularity for the Signorini problem and its free boundary
| journal = [[Invent. Math.]]
| language = English
| series =
| volume = 1
| pages = 1–82
| year = 2016
| issue = 1
| doi = 10.1007/s00222-015-0608-6
| arxiv = 1310.2511
| bibcode = 2016InMat.204....1A
| zbl =
}}.
== External links ==
*{{springer
| title= Signorini problem
| id= S/s110130
| last= Barbu
| first= V.
}}
* [https://www.scilag.net/problem/G-180630.1 Alessio Figalli, On global homogeneous solutions to the Signorini problem],
{{DEFAULTSORT:Signorini Problem}}
[[Category:Calculus of variations]]
[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Partial differential equations]]